Lectures on probability theory and statistics: Ecole d'Ete de Probabilites de Saint-Flour XXIX - 1999

Lecture Notes in Mathematics Editors: J.–M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1781

3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo

E. Bolthausen

E. Perkins

A. van der Vaart

Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXIX - 1999 Editor: Pierre Bernard

13

Authors Erwin Bolthausen Inst. of Mathematics University of Zurich 8057 Zurich, Switzerland e-mail: [email protected]

Aad van der Vaart Dept. of Mathematics Free University De Boelelaan 1081a 1081 HV Amsterdam, Netherlands e-mail: [email protected]

Edwin Perkins Dept. of Mathematics The University of British Columbia sc121-1984 Mathematics Road Vancouver scBC V6T 1Z2, Canada e-mail: [email protected]

Editor Pierre Bernard Laboratoire de Math´ematiques Appliqu´ees UMR CNRS 6620, Universit´e Blaise Pascal Clermont-Ferrand, 63177 Aubi`ere Cedex, France e-mail: [email protected]

Cover: Blaise Pascal (1623-1662) Cataloging-in-Publication Data available Die Deutsche Bibliothek - CIP-Einheitsaufnahme Lectures on probability theory and statistics / Ecole d’Eté de Probabilités de Saint-Flour XXIX - 1999. E. Bolthausen ; E. Perkins ; A. van der Vaart. Ed.: Pierre Bernard. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; Vol. 1781) ISBN 3-540-43736-3 Mathematics Subject Classification (2000): 60-01, 60-06, 60F05, 60F10, 60G57, 60J15, 60K35, 62-01, 62-06, 62F12, 61G05, 62G20, 82B24, 82B26 ISSN 0075-8434 Lecture Notes of Mathematics ISSN o721-5363 Ecole d’Et´e de Probabilit´es de St. Flour ISBN 3-540-43736-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10866547

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Table of Contents

Part I Erwin Bolthausen: Large Deviations and Interacting Random Walks 1

On the construction of the three-dimensional polymer measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2

Self-attracting random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3

One-dimensional pinning-depinning transitions . . . . . . . . . . . 105

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Edwin Perkins: Dawson–Watanabe Superprocesses and Measure-valued Diffusions

Part II

I

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

II

Branching Particle Systems and Dawson–Watanabe Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

III Sample Path Properties of Superprocesses . . . . . . . . . . . . . . . . 193 IV Interactive Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 V

Spatial Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Part III

Aad van der Vaart:

Semiparametric Statistics

1

Lecture: Introduction, Tangent Sets . . . . . . . . . . . . . . . . . . . . . . 336

2

Lecture: Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

VIIIººººººTableºofºContents

3

Lecture: Calculus of Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

4

Lecture: Gaussian Approximations . . . . . . . . . . . . . . . . . . . . . . . . 370

5

Lecture: Empirical Processes and Consistency of Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

6

Lecture: Empirical Processes and Normality of Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

7

Lecture: Efficient Score and One-step Estimators . . . . . . . . . 412

8

Lecture: Rates of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

9

Lecture: Maximum and Profile Likelihood . . . . . . . . . . . . . . . . 433

10 Lecture: Infinite-dimensional Z-Estimators . . . . . . . . . . . . . . . . 446 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Part I

Erwin Bolthausen:

Large Deviations and Interacting Random Walks

E. Bolthausen, E. Perkins, A. van der Vaart: LNM 1781, P. Bernard (Ed.), pp. 1–124, 2002. c Springer-Verlag Berlin Heidelberg 2002


Table of Contents

1

On the construction of the three-dimensional polymer measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The skeleton inequalities and boundedness properties . . . . . . . 14 ε , ε → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.3 The convergence of PT,β

2

Self-attracting random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A maximum entropy principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The diffusive phase for self-attracting random walks . . . . . . . . 2.4 The collapsed phase for self-attracting random walks . . . . . . . . 2.4.1 The tube property: Proof of Proposition 2.35 . . . . . . . . 2.4.2 Tightness: Proof of Proposition 2.37 . . . . . . . . . . . . . . . . 2.4.3 Proof of Theorem 2.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 A droplet construction for the Wiener sausage . . . . . . . . . . . . . 2.6 Moderate deviation for the Wiener sausage . . . . . . . . . . . . . . . . 2.6.1 Introduction and heuristics . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Analytical properties of the variational problem . . . . . 2.6.3 Proof of the upper bound in Theorem 2.54 . . . . . . . . . . 2.7 Remarks on the polaron problem . . . . . . . . . . . . . . . . . . . . . . . . .

3

One-dimensional pinning-depinning transitions . . . . . . . . . . . 105 3.1 Wetting transition for a one dimensional random walk . . . . . . 105 3.2 A heteropolymer near an interface . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

39 39 44 51 55 56 61 69 72 80 80 85 89 101

Introduction The lectures presented here treat three closely related topics on random walks with self-interactions or with interactions with a wall. In some cases, the “random walk” is a Brownian motion. All the topics have versions for random walks and for Brownian motion, but not in all cases both versions have been proved. The first topic addressed is the so called polymer measure in three dimension, also called Edwards’ model. This is a construction of a Brownian motion with local self-repellence, which is given by a δ-function. There are considerable difficulties to define this rigorous for dimensions larger than 1. The interaction in terms of this δ-function is not defined at all. One then tries to work with a regularized version, for instance smoothing the δ-function, and removes the regularization in a limiting procedure, proving that the limit measure exist. The first construction in the difficult three-dimensional case was by Westwater who in two celebrated papers in the early eighties proved that a suitably regularized version converges if the regularization is taken away. The two-dimensional case is easier and had been treated earlier by Varadhan. Westwater’s approach is extremely complicated, and essentially nobody seems to haven taken the pains to study his papers and his methods. Not much later, there has been an alternative approach, first in the context of quantum field theory, by Brydges, Fr¨ ohlich and Sokal, and a bit later also for the polymer problem by Bovier, Felder and Fr¨ ohlich. Their approach is much simpler, but also had considerable shortcomings. The main one was that it was impossible to speak of the polymer measure, as the proof gave only boundedness properties of finite dimensional distributions of (lattice) regularized versions, from which the existence of convergent subsequences could be derived. With such a method, it is then difficult to prove important properties of the measure. In a paper of mine in 1992, most of these shortcomings have been removed, and the convergence of a regularized version has been proved. The topic was vigorously taken up by X.Y. Zhou who wrote a number of papers (mostly with Albeverio, and one with me), extending the approach for instance to arbitrary coupling constants, identifying the measure with the one constructed by Westwater, and proving limit theorems for self-repellent discrete random walks converging to the measure. It was an extremely sad event when Zhou died suddenly in 1996. The second chapter will deal with self-attracting random walks. The selfattraction is given, too, in terms of a Gibbsian description, which contrasts with some models investigated recently in the probability community. The most natural example would be to change the sign of the coupling constant in the standard (weakly) self-avoiding case, but it is easy to see that this is not an interesting object as the attraction would be far too strong. So one is led to models with weaker interaction, namely where the coupling constant decays in time. Somewhat surprisingly, this model has a collapse transition in two and more dimensions, changing from a collapsed state when the coupling

6

Introduction

constant is large, to a diffusive one for weak couplings. The diffusive phase has been studied by Brydges and Slade. The collapsed state is investigated in a paper of Uwe Schmock and myself. Collapse transitions are well known in statistical mechanics in many models with self-attraction, but in most cases, there is no rigorous proof. A part of the second chapter will be devoted to problems around the Wiener sausage, where closely related effects appear. A self-attracting path measure (starting for instance with the Brownian motion) is obtained by transforming the measure by favoring paths with small Wiener sausage. It turns out that the path measure obtained in this way, leads to a kind of droplet construction, where the droplet describes in which region of the space the paths have to concentrate under the new measure. This droplet is somewhat trivial, being just a ball, a fact which is related to the standard isoperimetric problem. Recently, after previous work in the two-dimensional case by Sznitman and by myself, Povel has been able to prove in any dimension that the droplet concentrates in L∞ near the optimal droplet. The behavior of this model also depends crucially on the coupling constant chosen. It turns out that a model with decaying coupling constant is just diffusive if the decay is too fast. There is a critical case where the “droplet picture” starts to dissolve, which is quite interesting, and has recently been investigated by M. van den Berg, F. den Hollander and me. I will present the main ideas on this topic. The second chapter circles around models which have a localizationdelocalization transition, and this topic is continued in the last chapter which discusses two models with localization-delocalization phenomena of a somewhat different kind, namely coming from an interaction of a random walk with a “wall”. The chapter covers a model of a so called hetero- or copolymer with a localization-delocalization phase transition, and furthermore a so-called wetting transition in dimension one. Some comments about the degree in which technical details will be given in these lectures. Some of the proofs presented here would be technically very lengthy if given in all details. For instance, a full and complete proof of the construction of the three dimensional polymer measure would still require considerable space, but in fact, some of the calculations and estimates are quite repetitive, and it would only be tiring if all of them would be presented. As a rule, I am trying to present for most of the results some of the very core arguments in details. I will furthermore essentially concentrate on the probabilistic aspects, just citing the analytic ones. Many parts of these lectures can be read independently. In particular, the first chapter on the three-dimensional polymer measure stands somewhat apart. The other parts are all closely connected with large deviations (Chapter 1 actually, too, but somewhat hidden).

1. On the construction of the three-dimensional polymer measure

1.1 Introduction An outstanding open problem in probability theory is the determination of the mean end to end distance of a standard self-avoiding random walk on the d-dimensional lattice Zd for d = 2, 3 (and 4). Given n ∈ N, let Ωn be the set of paths ω of length n: def

Ωn = {ω = (ω0 , ω1 , . . . , ωn ) : ωi ∈ Zd , ω0 = 0, |ωi − ωi−1 | = 1 for 1 ≤ i ≤ n}, and the set of self-avoiding paths def

ΩnSA = {ω ∈ Ωn : ωi = ωj for i = j}. The main problem is to derive precise information about the asymptotic behavior of |ΩnSA |, the number of self-avoiding paths, and about the mean length of self-avoiding paths:
def ωn SA = ωn /|ΩnSA | SA ω∈Ωn

where   is the Euclidean length. From arguments in theoretical physics (conformal field theory, expansion techniques) it is believed that ωn SA scales with n3/4 for d = 2, and with nν with ν slightly less than 3/5 for d = 3. Also, the scaling limits, i.e. the asymptotic distribution of ωn /ωn SA should be non Gaussian (see [52]). From dimension 4 onwards, the scaling limits are becoming Gaussian, with a slight correction to ordinary central √ √ limit scaling for d = 4, where ωn SA is believed to be of order n 8 log n. The case of d ≥ 5 is completely settled: Starting with work by Brydges and Spencer [25] who introduced the lace expansion, and culminating with Hara and Slade [55]. An excellent monograph on these and related topics is [59]. (For a recent conceptually simple approach, see [18]). There is no (published) proof for d = 4 which is not (directly) tractable by lace expansions (see [22], [58], [26] for partial results). I will not give any discussion of these techniques here. One of the results I discuss is a very weakly interactive case for d = 3, where the interaction

8

1 On the construction of the three-dimensional polymer measure

is so weak that one has ordinary scaling, but where nevertheless the scaling limit of (ω[nt] /ωn )0≤t≤1 , which is shown to exist, is not Gaussian, but instead given as the so called Edwards’ model first constructed rigorously by Westwater [76]. We now introduce the so called weakly self-avoiding random walks. Here, all paths in Ωn receive positive weight, but the ones with many intersections are “punished”. This is achieved by choosing a parameter λ ∈ (0, 1). Then every path ω ∈ Ωn gets its relative weight decreased by a factor (1 − λ) for every self intersection, i.e. we define the probability measure on Ωn by def Pˆn,λ (ω) = def

where Zn,λ =



Π

Π

(1 − λ1ωi =ωj ) / Zn,λ ,

0≤i<j≤n

(1 − λ1ωi =ωj ) .

ω∈Ωn 0≤i<j≤n

(Remark: Through these notes, we will always use Pˆ to denote measures on path spaces obtained from “simple” random walk measures by introducing interactions, self-repelling in this chapter, and self-attracting in the next.) We rewrite the above measure by setting (with a slight abuse of notation)   n
β Pˆn,β (ω) = exp − 1ω =ω  Zn,β , (1.1) 2 i,j=1 i j where β = − log(1 − λ) ∈ (0, ∞), and Zn,β being the appropriate norming. Remark that the diagonal part in the summation is cancelling. We can also rewrite the interaction: n


1ωi =ωj =

i,j=1




n (x, ω)2 ,

x∈Zd

where n (x, ω) is the discrete local time def

n (x, ω) =

n


1ωj =x .

j=0

This is the so called Domb-Joyce model. The above expression for the Domb-Joyce model naturally leads to the question if similar models exist starting with the Brownian motion instead of the random walk and how the relations between this and the discrete models are. We start with the Wiener measure PT on C0d (T ), the set of continuous paths ω : [0, T ] → Rd , starting at 0, and we want to define the polymer measure formally by 

T T 1 β PˆT,β (dω) = exp − dt ds δ(ωt − ωs ) PT (dω), (1.2) ZT,β 2 0 0

1.1 Introduction

9

where δ is the Dirac function. There is evidently some trouble defining this, as the formal expression



T



T

dt δ(ωt − ws ) =

ds 0

dx T (x, ω)2 ,

0

where



T

δ(ωs − x)ds,

T (x, ω) = 0

i.e. the L2 -norm of the local time, only makes sense for d = 1. The trouble is also revealed by formally calculating the expectation under Wiener measure



T

dt δ(ωt − ωs ) = 2

ds

ET 0



T

0

dsdt pt−s (0),

(1.3)

0≤s≤t≤T

where pu (x) is the transition density of Brownian motion, i.e. pu (x) = (2πu)−d/2 e− def


x
2 2u

.

However, the right hand side of (1.3) is evidently divergent for d ≥ 2. There are a number of ways in which one can try to remedy the situation. The first idea, but not the easiest one, is to step back to the Domb-Joyce model and to try to make some limiting procedures with the lattice spacing going to 0, and an appropriate dependence of β on n. This is possible, but is somewhat delicate, and has only recently been done in a completely satisfactory way [1] for d = 3. I will discuss that below. Another approach is to replace δ by a smoothed version, e.g. pε , ε > 0, and then let ε → 0. For d = 2 this was the way in which Varadhan proved the existence of the polymer measure (1.2). The most convenient way however is to use some gap regularization. Observe that the right hand side of (1.3) is divergent only because of the integration near the diagonal. If we leave a gap between s, t, e.g. integrating only over s + ε ≤ t, ε > 0, then this stays finite. It is in fact known that



T −ε

ε (ω) = J0,T

T

dt δ(ωt − ωs )

ds s+ε

0

is well defined, PT - a.s. As this is still only a formal expression, some comments are in order. We can define, for every a > 0, ε,a (ω) = J0,T



T −ε

T

dt pa (ωt − ωs ),

ds 0

s+ε

and then (with fixed ε > 0) let a → 0. This limit exists e.g. in L2 (see [64]), ε . The limit has nice properties, e.g. it is a.s. and is what we denote by J0,T

10

1 On the construction of the three-dimensional polymer measure

continuous in ε, T . We will not go into a discussion of these properties, but simply refer to the relevant literature, e.g. [64]. We then define our regularized Edwards’ model by ε ε PˆT,β (dω) = exp(−βJ0,T (ω))PT (dω)/ZT,β,ε .

(1.4)

Theorem 1.1. For T, β > 0, d = 2, 3, the limit ε PˆT,β = lim PˆT,β ε→0

exists as a weak limit of probability measures on Cod (T ). We will focus on the case d = 3 which is considerably more delicate than the case d = 2. The theorem is essentially due to Westwater [76]. The only difference is that he took a slightly different gap regularization. The procedure Westwater follows is, however, extremely difficult. We give some comments on it below. We explain here an approach which is much easier and is based on so called skeleton inequalities. This method had been introduced by Brydges, Fr¨ ohlich and Sokal [23] in Euclidean ϕ4d quantum field theory and had then been adapted to the polymer problem in [19]. From a probabilistic point of view, the results in [19] have however a number of shortcomings. The most serious one is that no convergence is proved, but only boundedness properties which made it possible to prove the existence of convergence subsequences. There was then no possibility for an identification of the process for instance with the one constructed by Westwater. The above authors also had used a lattice regularization, and it was not even clear that there are limits which are rotational symmetric. Another point is that the results are formulated in terms of Laplace transforms in time, and the problem how to get polymer measures with fixed time horizon had not been addressed. For this reason, the approach was generally thought to be simple but that it would give only somewhat weak results. However, at least for the polymer case, some of the shortcomings can be remedied, and a modification proving convergence at fixed time has been developed in [9]. This is explained below. In [9], the polymer measure was constructed only for a small coupling parameter β. This restriction has later been removed in [2], by a simple but clever argument, which we will include here, too. First some comments on the approach by Westwater [76]. He uses no regularization of the δ-function but a slightly different gap regularization than that explained above. Take T = 1 (for notational simplicity). Then



1/2

1

dt δ(ωt − ωs )

ds

X0 (ω) = 0

1/2

is P1 -a.s. well defined (in the sense described above). On the next level, one defines

1.1 Introduction





1/4

X1,1 (ω) =

1/2

dt δ(ωt − ωs ),

ds 0



11

1/4



3/4

X1,2 (ω) =

1

dt δ(ωs − ωt ),

ds 1/2

3/4

and then of course Xn,i (ω) =



(2i−1)2−n−1

i2−n

ds (i−1)2−n

(2i−1)2−n−1

dt δ(ωs − ωt ),

n ≥ 1, 1 ≤ i ≤ 2n . These variables have a number of simple properties. For fixed n, the 2n variables Xn,i are evidently independent. Furthermore,√the law of Xn,i by simple Brownian rescaling is the same as that of Xn−1,i / 2, n ≥ 2 for d = 3. The main difficulty is that for different n, the Xn,i are not independent. Westwater proves that there is a near independence between Xn,i and Xm,i if |m − n| is large. In other words, there is near independence between short and long range self intersections. Westwater then proves, using N,W W exists where this property, that lim Pˆ1,β N →∞



N,W W Pˆ1,β (dω) = exp −β



n

2 N



Xn,j (ω) P1 (dω)/ZβN .

n=0 j=1

The picture below (Fig. 1.1) shows the domain of integration for N = 2. The main disadvantage of the Westwater approach is that it is extremely complicated which is mainly due to the fact that it makes bad use of the fact that Xn,i ≥ 0. A further enormous complication arises because the Xn,i do not have exponential moments. It has recently been proved by Albeverio and Zhou [2] that the Westwater process coincides with the one of Theorem 1.1. This might look obvious, but in fact, the removal of the gap is quite subtle as will become apparent. One of the motives to investigate the continuous polymer measures had certainly been the hope that they shed some light on the discrete model. The relation is however quite delicate. To see what the appropriate scaling should be, we will perform some formal calculations. We consider the polymer measure on a time slot [0, T ] with a coupling parameter βT > 0 which may depend on T . Formally 

T T ˆ ds dt δ(ωt − ωs ) PT (dω)/Z. dPT,β = exp −βT T

0

Performing Brownian scaling

s

√ ω ˜ t = ωtT / T , t ≤ 1,

and using

12

1 On the construction of the three-dimensional polymer measure

t

s T Fig. 1.1.





T

dt δ(ωt − ωs ) = T

ds 0

T

s

2− d 2





1

1

dt δ(˜ ωs − ω ˜t)

ds 0

s

we see that for βT = βT 2 −2 , the distribution of the rescaled path under PˆT,βT is just Pˆ1,β . (This is of course not a rigorous proof, but the statement is correct). Anyway, this suggests that starting with a standard random walk (ω0 = 0, ω1 , . . . , ωT ) on Zd , and the weakly self-avoiding walk  
1 P˜T,β (ω) = exp −β 1ωi =ωj  . ZT,β d

0≤i<j≤T

one has Theorem 1.2. Assume d ≤ 3 and β > 0. Then lim P˜T,βT −2+d/2 YT−1 = Pˆ1,β

T →∞

√ where YT : ΩT → Cod (1) is defined by YT (ω)(i/T ) = ωi / T , and linearly interpolated between.

1.1 Introduction

13

The above Theorem is easy for d = 1, has been proved by Stoll [69] for d = 2, and in [1] for d = 3. It is of course far from the “real” question, namely what happens with P˜T,β for fixed β as T → ∞. On the other hand, even in the above “very weakly” self-avoiding case, the limiting measure for d = 3 is singular with respect to any Wiener measure, and is non-Gaussian, as has been proved by Westwater. Remark that in the two dimensional case, the T -dependence of βT is βT = β/T . This will be important in Chapter II. There are considerable technical difficulties to prove Theorem 1.2 for d = 3. The main problem is to show that the short range intersections, where the random walk does not quite look like a Brownian motion, do not disturb the limiting picture. We will not give a detailed proof here. It is essentially a modification of the arguments in the proof of Theorem 1.1 but requires some additional nontrivial arguments. It is to be expected that the limiting behavior of the weakly self-avoiding model (i.e. P˜T,β for fixed β, T → ∞) is by the above rescaling related to the β → ∞ behavior of the polymer measure Pˆ1,β . There is no proof of this for d ≥ 2. Even the d = 1 case is very far from trivial, and has only recently been solved by van der Hofstad and den Hollander [57]. We give an outline of the rest of this chapter. We entirely focus on d = 3 which is the most delicate case. In Section 1.2 we discuss the boundedness properties of the so called two point functions. This follows closely the approach in [23] and [19], but there are some differences. First, we avoid using Laplace transforms in time. Proving things in Laplace transformed versions is technically simpler but then one has the trouble to invert the result. This inversion is not done in the above mentioned papers of Brydges, Fr¨ ohlich, Sokal and Bovier, Felder and Fr¨ ohlich. We also derive relatively sharp pointwise estimates (in contrast to Lp -estimates). I believe that in the long run, methods for fixed time and directly in xspace give sharper results and are more transparent than when using various transforms which have to be inverted. Another example of this is a recent direct approach to weakly self-avoiding walks for d ≥ 5 [18], which is conceptually much simpler than older ones and yields somewhat sharper results. I will present some details of the proof of Theorem 1.1, but not all. First, I take some continuity properties of intersection local times for granted. These are modifications of classical results proved by Rosen. For details I refer to the Appendix of [9]. The basic inequalities are explained in details but the calculations are somewhat repetitive and I will not give all of them. The boundedness properties immediately imply the tightness of the measures, as ε → 0. With the inequalities derived in 1.2, it is however not possible to prove convergence. In Section 1.3, we derive some alternative inequalities which are more delicate to handle, but with which it is possible to prove convergence. The approach in [19] and [9] had originally been purely perturbative, but by an observation of [2], this can be extended to arbitrary β > 0.

14

1 On the construction of the three-dimensional polymer measure

1.2 The skeleton inequalities and boundedness properties Let ε > 0, 0 ≤ t < t + ε ≤ T < ∞, and set T T −ε def ε Jt,T (ω) = ds1 ds2 δ(ωs1 − ωs2 ) t

s1 +ε

which can be defined as the a.s. limit T −ε T ds1 ds2 pa (ωs1 − ωs2 ). lim a↓0

t

s1 +ε

For T − t ≤ ε, we put = 0. The existence of this limit can be proved by Fourier techniques (see [64]). We will below perform some formal manipulations with δ-functions, which all can easily been justified (for a fixed ε-gap) by replacing δ by pa and letting a → 0. All the serious trouble is coming when discussing the ε → 0 limit, and we will focus on that. We consider the so called two point functions g εT,β (x) defined to be the density of the measure  ε ET exp(−βJ0,T ); ωT ∈ dx ε Jt,T (ω)

on Rd . It is convenient to write this formally as  ε g εT,β (x) = ET exp(−βJ0,T )δ(ωT − x) . We write g because these quantities have to be slightly modified later on, and we will switch then to g. Evidently, we have for 0 ≤ t ≤ T  ε pt ∗ g εT −t,β = ET exp(−βJt,T )δ(ωT − x) . Setting t = 0 gives g T and t = T gives pT . By the fundamental theorem of calculus, we therefore arrive at T d ε pT (x) − g T,β (x) = dt (pt ∗ g εT −t,β )(x) dt 0

  T d ε ε dt ET −β )δ(ωT − x) . = Jt,T exp(−βJt,T dt 0  T d ε Now, dt Jt,T (ω) = − t+ε ds δ(ωs − ωt ), if t ≤ T − ε, and 0 otherwise, and we therefore get pt (x) −

g εT,β (x)

=β



T −ε

T ε ds E(δ(ωs − ωt ) exp(−βJt,T )δ(ωt − x)).

dt 0

t+ε

(1.5)

1.2 The skeleton inequalities and boundedness properties

15

The manipulation may look somewhat cavalier, in particular as the intersection local times are not differentiable in the time limits, but they are easily justified. We will derive some concrete inequalities involving g. These inequalities are the only objects we are interested in. These inequalities do make sense also when δ is replaced by pa in which case all the manipulations are easily justified, and we can take the a → 0 limit in the end. We will however stick to the δ notation which is evidently more convenient. We will often drop ε, β in the notations but they should be remembered to be present. On the right hand side of (1.5), we can split the interaction on [t, T ] into the self-interactions on [t, s] and [s, T ] and the interactions between these intervals: ε ε ε ε Jt,T = Jt,s + Js,T + Jt,s;s,T ,

(1.6)

where ds1 ds2 δ(ωs1 − ωs2 ).

ε Jt,s;s,T =

(1.7)

t≤s1 ≤s≤s2 ≤T s2 −s1 ≥ε

In (1.5), there is no interaction inside the interval [0, t], and also none between this interval and the next. However, there is an interaction left between [t, s] and [s, T ], which is given by the third summand on the right hand side of (1.6). Without this interaction, the right hand side of (1.5) would just be



T −ε

dt 0



T

ds t+ε

dy pt (y)g s−t (0)g T −s (x − y).

(1.8)

(We have dropped β, ε in g for notational convenience.) It is convenient to introduce a diagram notation for this and more complicated expressions. The so-called free propagator is

and the propagator with interaction is

One should however keep in mind that there is always an ε present: otherwise this interactive propagator is not defined (yet). We also always define g T (x) = 0 for T < ε. We can then write the expression (1.8) as

16

1 On the construction of the three-dimensional polymer measure s−t

dt ds

T −s

t

0

x

s−t

0≤t≤s≤T

dt ds

=

0

T −s

t

x

×

0

0≤t≤s≤T

where the · means that we take convolution in x-space (i.e. in R3 ). The main trouble is evidently coming from the presence of the interaction summand of (1.6). This is now handled by some simple inequalities which use the fact that this interaction term is non-negative. We therefore get the two inequalities ε e−βJt,s;s,T ≥ 1 − βJt,s;s,T , ε

(1.9)

and β2 ε (J )2 . (1.10) 2 t,s;s,T Implementing the second summand in the right hand side of (1.5) gives a contribution ε e−βJt,s;s,T ≤ 1 − βJt,s;s,T + ε

−β

2

ds1 ds2 ds3 ds4

A(ε)

dy

dz ps1 (y)

 ε ε × E e−β (Js1 ,s3 +Js3 ,T ) δ(ωs1 − y)δ(ωss2 − z)  × δ(ωs3 − y)δ(ωs4 − z)δ(ωT − x) ,

(1.11)

where A(ε) = {(s1 , s2 , s3 , s4 ) : 0 ≤ s1 ≤ s2 ≤ s3 ≤ s4 ≤ T, s3 − s1 ≥ ε, s4 − s2 ≥ ε}. It should now be observed that Jsε1 ,s3 still contains all the interaction on the interval [s1 , s3 ] and similarly for Jsε3 ,T . It looks that we have gained nothing as things are becoming more and more complicated. However, dropping the remaining interaction between the intervals [s1 , s2 ] and [s2 , s3 ], and between [s3 , s4 ] and [s4 , T ] gives an estimate in the right direction if we use this together with (1.9), simply because Jsε1 ,s3 ≥ Jsε1 ,s2 + Jsε2 ,s3 , Jsε3 ,T ≥ Jsε3 ,s4 + Jsε4 ,T . As the remaining propagator will be crucial, we give it a new name: ε GT (x) = ds1 ds2 g s1 (x)g s2 −s1 (x)g T −s2 (x). (1.12) 0≤s1 ≤s2 ≤T

s2 ≥ε,T −s1 ≥ε

1.2 The skeleton inequalities and boundedness properties

17

In diagram notation, we have s1



ε

GT (x) =

s2 − s1

0

ds1 ds2

x

T − s3

0≤s1 ≤s2 ≤T s2 ≥ε,T −s1 ≥ε

As this propagator is quite crucial, we introduce a new notation for it: T

0

def

x

=

ε

GT (x)

We will also need the corresponding propagator where the g are replaced by the free propagator p, but where the ε-restrictions on the integration are kept. This is denoted by def



PTε (x) =

ds1 ds2 ps1 (x)ps2 −s1 (x)pT −s2 (x), 0≤s1 ≤s2 ≤T

s2 ≥ε,T −s1 ≥ε

for which we use the diagram notation T

0

x

def

=

PTε (x)

PTε (x) is not defined for x = 0 as, even with the gap-condition, the integral is divergent. However, for x = 0, the integrals are perfectly convergent. In order to recast the first inequality of (1.9), we still have to look at the contribution of 1. Implementing this part into (1.5) just means that we forget about the interaction between [t, s] and [s, t]. We therefore get our first basic inequality:

(0

T

T

(0

x)

≥β

ds1 ds2

s2 − s1

x) T − s2

s1

0

x ×

(1.13)

0

0≤s1 ≤s2 ≤T

−β 2



0≤s1 ≤s2 ≤T

ds1 ds2

0

s2

s2 − s1

T − s2

x

18

1 On the construction of the three-dimensional polymer measure

To get an upper bound, we have to expand the interaction between the legs in (1.11) in the same way as before, and we have also to take into account the third summand of the second inequality in (1.10). The reader will convince himself quickly that all the contributions are of the same form, namely

A3 (ε)

ds (ps1 ∗ [g ∆s3 ((g ∆s1 g ∆s2 ) ∗ (g ∆s4 g ∆s5 ))] ∗ g ∆s6 )(x) =

s1

0

ds A3 (ε)

∆s1 ∆s2

(1.14)

∆s4 ∆s3

∆s5

∆s6

x

where A3 (ε) is some subset of {s = (s1 , s2 , . . . , s6 ) : 0 ≤ s1 ≤ s2 ≤ . . . ≤ s6 ≤ T } with a number of ε-gap conditions, whose exact form will be no longer of any importance, and ∆si = si+1 − si (s7 = T ). We will stick to this notation of ∆si also in cases where the number m of different si is not 6, always putting ∆sm = T − sm . The above contribution pops up from multiplying out the square of the third summand in (1.10), dropping afterwards all the remaining interactions between the time slots there, and by expanding the interaction between the time slots in (1.11) once. The inequalities evidently all go in the correct direction to yield

(0

T

x)

T

(0

x)

≤ β

ds 0

0≤s1 ≤s2 ≤T

∆s1

s1

∆s2



− β2

ds 0

x ×

0

(1.15)

∆s1

s1

∆s2

x

0≤s1 ≤s2 ≤T

+ 4β 3

ds

0

s1

∆s1 ∆s2

∆s4 ∆s3

∆s5

∆s6

x,

0≤s1 ≤···≤s6 ≤T

It is worthwhile to pause and contemplate if anything has been achieved with the inequalities (1.13) and (1.15). A moment’s reflection reveals that this is not the case. For instance, the first diagram contains an integration over a ≥ ε. If for the moment, we let drop the interaction “loop” g s (0) over a time  completely, we have ps (0)ds which is divergent as ε → 0. One might think ε  that the interaction could help and g s (0)ds would be convergent, but this ε

1.2 The skeleton inequalities and boundedness properties

19

is not the case. For similar reasons, the second summand is divergent for ε → 0 (but actually only marginally). We are more fortunate with the third summand on the right hand side of (1.15). If we drop all the interactions we arrive at ds (ps1 ∗ [p∆s3 ((p∆s1 p∆s2 ) ∗ (p∆s4 p∆s5 ))] ∗ p∆s6 )(x), A3 (ε)

and if we drop also all the gap conditions (this gives an estimate from above), we arrive at ds{ps1 ∗ [p∆s3 ((p∆s1 p∆s2 ) ∗ (p∆s4 p∆s5 ))] ∗ p∆s6 }(x), 0≤s1 ≤s2 ≤s3 ≤s4 ≤s5 ≤s6 ≤T

and it is elementary to check that this is convergent! (For d = 3.) Therefore, there might be some hope that the third summand on the r.h.s. of (1.15) is o.k. The way to get also the first two summands right is to modify the definition of g slightly by introducing so-called counterterms which are cancelling these divergencies. We will then prove pointwise boundedness and decay properties by an appropriate recursion Ansatz, assuming β small enough. As these counterterms are supposed to cancel the loop and “three leg” divergency, we define them by the corresponding objects for the free propagator: 1

ds ps (0)

κ1 (ε) =

(1.16)

ε

and κ2 (ε) = 0

=

1

ds Psε 1 ds1 ds2 ds3 dx ps1 (x)ps2 −s1 (x)ps3 −s1 (x),

0≤s1 ≤s2 ≤s3 ≤1

s2 ≥ε,s3 −s1 ≥ε

i.e.

0

κ 2 (ε) = 0≤s1 ≤s2 ≤s3 ≤1 s2 ≥ε,s3 −s1 ≥ε

s1 ∆s1 ∆s2

1

κ1 (ε) is of course just 2(2π)−3/2 ( √1ε − 1). κ2 is slightly more complicated to evaluate. Remark first that dx pt1 (x)pt2 (x)pt3 (x) = (2π)−3 [t1 t2 + t1 t3 + t2 t3 ]−3/2 ,

20

1 On the construction of the three-dimensional polymer measure

and therefore −3





1

κ2 (ε) = (2π)

dt1

1

1

3ε

= (2π)−3



dt2 [(t1 + t3 )t2 + t1 t3 ]−3/2 + O (1)

0

0

= (2π)−3

1

dt3

0

= (2π)−3



1

1

3ε

du u du u du u

(ε−t1 )∨(ε−t3 )∨0



u

0



2dv  + O (1) v(u − v) + u ((ε − v) ∨ (ε − v + u))+

u−ε

2dv  + O (1) v(u − v) + u ((ε − v) ∨ (ε − v + u))+

ε



u

0

2dv  + O (1) = (2π)−2 | log ε| + O (1) . v(u − v)

We therefore see that this is just barely divergent. The divergence of κ2 (ε) is actually making all the trouble for d = 3. It should be remarked that ε and κ2 (ε) the variance. If the variance stays κ1 (ε) is essentially just EJ0,1 bounded as ε → 0, one can apply what in quantum field theory is called vacuum renormalization, i.e. one just replaces J by J −EJ getting something which is convergent. This is the approach of Varadhan for d = 2 [75]. The renormalized interaction is now just ε,β ε = βJt,T − β(T − t)κ1 (ε) + β 2 (T − t)κ2 (ε). Rt,T

It is important that the time enters linearly into the counterterms. We put def

ε,β ε (x) = ET (exp(−R0,T )δ(ωT − x)). gT,β

We again apply

T

ε pT (x) − gT,β (x) =

dt 0

d ε,β E(exp(−Rt,T )δ(ωT − x)). dt

There are only small changes to our inequalities (1.13) and (1.15). The presence of the counterterms gives only the additional summand



2

(−βκ1 (ε) + β κ2 (ε)) =

−β

1 ε

ds

×

0 s

0

(1.17)

dt pt ∗ gT −t (x)

+ β2

T

dt 0

T

0



s

1

ds 0

t

0

1

T −t

x

1.2 The skeleton inequalities and boundedness properties

21

in both cases. We don’t change our diagram notations, but from now on, the propagators with interactions always refer to g and not to g, i.e. to the propagators including the counterterms. The main pointwise estimate for small β > 0 is Proposition 1.3. There exists C > 0 and β0 > 0, such that for all β ∈ [0, β0 ], T ≤ 1, x ∈ R3 , ε > 0 one has √ ε (x)| ≤ Cβ T p2T (x). |pT (x) − gT,β The proof of this estimate is by a recursion argument using our basic inequalities (1.13) and (1.15) with the appropriate corrections coming from the counterterms, i.e. (1.17). We set K0 (ε, β) = sup

0
ε |gT,β (x) − pT (x)| √ . T p2T (x) x∈R3

sup

Remark first, that K0 (ε, β) is finite for fixed ε > 0, β ≥ 0. In fact, for T ≤ ε, the interaction is 0, so ε (x) = pT (x) exp(βT κ1 (ε) − β 2 T κ2 (ε)), gT,β

so the sup over 0 < T ≤ ε is certainly finite, as pT (x) decays faster at |x| ∼ ∞ than p2T (x). For the same reason, the supremum is also finite on ε ≤ T ≤ 1. K0 (ε, β) looks being the right quantity for Proposition 1.3, but for technical reasons, we have to slightly change it, and we set  1    ε (0))ds . K(ε, β) = K0 (ε, β) ∨  (ps (0) − gs,β 0

The reason is that in order to estimate K0 , one has to use estimates on (ps − gs )ds. Evidently, this quantity itself cannot be controlled by K0 . This 0

is a slightly awkward point, and for that reason we have to work with K instead of K0 . The main work for proving Proposition 1.3 is then contained in Proposition 1.4. There exists a polynomial φ(x) with nonnegative coefficients, such that for all ε > 0, β ∈ [0, 1], one has K(ε, β) ≤ β φ(K(ε, β)). The proof of this is a bit lengthy and tedious but essentially rather straightforward. We give details of some parts of the estimates, namely the ones involving the divergent “three leg” diagram. In the next section where we prove convergence, we then focus on the other divergent part. Before we begin with that, we show how Proposition 1.4 implies Proposition 1.3. We still need a further result

22

1 On the construction of the three-dimensional polymer measure

Lemma 1.5. For any fixed β > 0, the function (0, 1)  ε → K(ε, β) is continuous. This follows well known techniques concerning intersection local times (see [64], and the Appendix of [9]). We will not give a proof here. The Proposition 1.4 and Lemma 1.5 imply Proposition 1.3 in the following way: Let def

(β) = inf{x ≥ 0 : x = βφ(x)}. If β is small enough then we have (β) ≤ cβ. We have K(ε, β) = 0 for ε = 1, and as K(ε, β) is continuous in ε > 0, it can never cross (β). We therefore get the estimate K(ε, β) ≤ cβ for all ε > 0 if β is small enough. This proves Proposition 1.3. To come now to the proof of Proposition 1.4, we get, using our inequalities (1.13) and (1.15) with the correction (1.17) from the counterterms:

|pT (x) − gT (x)| T

= (0

≤β

T

(0

x)

x) ∆s1

s1

ds 0

∆s2

x ×

0

0≤s1 ≤s2 ≤T

T − κ1 (ε)

T −s

∆s1

s1

ds 0

∆s2

0≤s1 ≤s2 ≤T

T − κ2 (ε)

s

ds 0

T −s

0

+ 4β

3

s1

ds 0

∆s1 ∆s2

∆s4 ∆s3

∆s5

∆s6

Here we have of course the modified definition s

0

(1.18)

x

0



+ β2

s

ds 0

x

def

=

Gs (x)

x

x

x

1.2 The skeleton inequalities and boundedness properties

23

but just without the bar, meaning that the appropriate counterterms are included. In the last summand of (1.18), the integration is over time slots for the vertices of the diagram which sum to T . We can drop the various ε-gap restrictions in that contribution, getting an upper bound. In contrast, the other two summands retain the gap restrictions. It is now fairly obvious, how the necessary cancellations take place. The counterterm κ1 (ε) is defined in such a way that it cancels the loop divergency, and κ2 (ε) is made such that it should cancel the “three leg” divergency. For the third summand, there is no further cancellation needed as it is convergent anyway. Of course, there is the problem that the times to not fit quite nicely, and in the second summand, the three leg divergency has to be “operated out” of the diagram and the remaining two ends “glued together”. We will not present all the estimates in details, but will show how to perform them for the second summand, which is the more delicate one. Lemma 1.6.

ds 0≤s1 ≤s2 ≤T

∆s1

s1

0 T

− κ2 (ε)

ds

0

s

∆s2

T −s

x

x

0

≤ φ(K) p2T (x)T 3/4 . Notations We use φ(x) as a generic polynomial with positive coefficients, not necessarily the same at different occurrences. K is always K(ε, β). We also use C as a generic positive constant, also not necessarily the same at different occurrences, which does not depend on ε, β. Remark 1.7.√The estimate is better than necessary for our estimate. The factor with T appears in the estimate of the first summand on the right hand side of (1.18). In order to prove the Lemma, we split things into three parts:

24

1 On the construction of the three-dimensional polymer measure

0

ds

∆s1

s1

0≤s1 ≤s2 ≤T

∆s2

T − κ2 (ε)

s

0

ds 0

≤

T −s

∆s1

s1

0

ds

x x ∆s1

−

∆s2

x

0≤s1 ≤s2 ≤T

+

0

ds

∆s1

ds

1

= I1 + I2 + I3 ,

∆s1

∆s2

1

x

T

s

s

ds 0

1

0

(1.19)

x

x×

s2

× 0

1 − ds

∆s2

∆s2

s2

− 0

+

∆s1

s1

T −s

x

0

say.

(Remember that Gs and Ps have the gap conditions and particularly are nonzero only if their time length is ≥ ε). First remark that for T ≤ 1, which we always assume, we have |gT (x)| ≤ pT (x) + |pT (x) − gT (x)| ≤ C (1 + K) p2T (x) .

(1.19)

The third summand I3 is very easy, we begin with that. The only difference between the two contributions inside is that in the first, the integration over ∆s1 is restricted to ε ≤ ∆s1 ≤ s2 . Therefore, using (1.19), we get



T

ds (ps ∗ gT −s )(x)

I3 =

Pu 1 du s

0



1

T

ds(ps ∗ gT −s )(x)| log(s)|

≤C 0



(1.20)

T

ds | log(s)| ≤ CT 3/4 p2T (x).

≤ C (1 + K) p2T (x) 0

Next, we estimate I1 , which is more complicated. We first split t1

0

t2 t3

t1

x − 0

t2 t3

x

1.2 The skeleton inequalities and boundedness properties

25

as a sum of expressions of the form ht1 (x)ht2 (x)ht3 (x), where ht is either pt (x) or gt (x) − pt (x), but where at least one of the h s is the latter. For definiteness, let us look at (gt1 (x) − pt1 (x))pt2 (x)pt3 (x) which we estimate in absolute value by √

t1 Kp2t1 (x)pt2 (x)pt3 (x) √ = C t1 K[2t1 t2 + 2t1 t3 + t2 t3 ]−3/2 pσ (x),

where def

σ =

2t1 t2 t3 . 2t1 t2 + 2t1 t3 + t2 t3

(1.21)

We also replace √ we again estimate √ g∆s2 in I1 by p∆s2 and the difference, where the latter by ∆s2 Kp2∆s2 (x). Evidently, the more of the ∆s terms we have, the better, so we look what happens if we just have one. Such a contribution to an upper bound of I1 is √

≤ CK

ds 0≤s1 ≤s2 ≤s3 ≤s4 ≤T s3 −s1 ≥ε,s4 −s2 ≥ε

def

∆s1 (ps1 +∆s4 ∗ pσ )(x) (2∆s1 ∆s2 + 2∆s1 ∆s3 + ∆s2 ∆s3 )3/2

(1.22)

def

where ds = ds1 ds2 ds3 ds4 , ∆si = si+1 −si and σ is from (1.21), ti replaced by ∆si . Of course, there are also summand with K 2 , K 3 , K 4 in the estimates (and correspondingly more ∆si ), but these can be estimated similarly. Let us look at what happens with the expression (1.22). Keeping ∆s1 , def

∆s2 ,∆s3 fixed and integrating over s1 gives just a factor t = T − (∆s1 + ∆s2 + ∆s3 ) = (s1 + ∆s4 ). Furthermore, a simple estimate for σ is σ ≤  i ∆si = T − t, which yields tpt (x) ≤ t−1/2 T 3/2 pT −σ (x). Therefore, √

(1.22) ≤ CKT

3/2

pT (x)

ti ≥0,




 ti <1

1−



t1 dt1 dt2 dt3

ti (t1 t2 + t1 t3 + t2 t3 )3/2

≤ CKT 3/2 pT (x). √ Remark that the t1 is doing the job of making the integral convergent. The other expressions get similar estimates, but we cannot always have pT (x).

26

1 On the construction of the three-dimensional polymer measure

However, p2T (x) is o.k., too, of K 2 , K 3 , K 4 . Altogether, we get

course. There are also expressions with

I1 ≤ φ(K)T 3/2 p2T (x),

(1.23)

which is much better than required. There remains to estimate I2 (which is the only place where we have to increase T to 2T in the estimate). Even if we drop all ε restrictions, Ptε (x) is evidently finite for x = 0, and has L1 -norm Pt0 1 = c/t. Despite the fact that this is divergent for t → 0, it is nevertheless true that for t ∼ 0, is Ptε (x) is essentially concentrated close to 0, and therefore there is not much difference between ps ∗ Ptε ∗ gu and (ps ∗ gu ) × Ptε 1 . If we set τ (s) = [∆s1 ∆s2 + ∆s1 ∆s3 + ∆s2 ∆s3 ]−3/2 , def

def

σ(s) = ∆s1 ∆s2 ∆s3 τ 2/3 , where as usual ∆si = si+1 − si , then p∆s1 (x)p∆s2 (x)p∆s3 (x) = (2π)−3 τ (s)pσ(s) (x).

Therefore I2 ≤ C

ds τ (s)(|ps1 +σ(s) − ps4 | ∗ gT −s4 )(x),

where the integration is over 0 ≤ s1 ≤ s2 ≤ s3 ≤ s4 ≤ T , and where we have dropped the ε-gap restrictions. We split the above integral in the part, where s1 + σ ≤ s4 /2 and the complement of this. On {s1 + σ ≤ s4 /2} there is actually no cancellation needed and we estimate |ps1 +σ − ps4 | ≤ ps1 +σ + ps4 , and of course, we also estimate  gT −s4 ≤ pT −s4 + K T − s4 p2T −2s4 ≤ φ(K)p2T −2s4 . Therefore, ds τ (s)(|ps1 +σ(s) − ps4 | ∗ gT −s4 )(x) {s1 +σ<s4 /2}



≤ φ(K)

ds τ (s)(p2T −s4 + p2T −2s4 +s1 +σ )(x) {s1 <s4 /2}



≤ φ(K)

ds τ (s)p2T (x) {s1 <s4 /2}



+ φ(K) {s1 <s4 /2,s4 >T /2}

ds τ (s)p2T −2s4 +s1 +σ (x).

1.2 The skeleton inequalities and boundedness properties

27

The integration of τ over s2 , s3 for given s1 , s4 is just Ps04 −s1 1 = c|s4 −s1 |−1 , and so the first summand is φ(K)T p2T (x). As for the second, it is 2 ≤ φ(K) ds τ (s)(T − s4 + s1 )−3/2 e−|x| /4T {s1 <s4 /2,s4 >T /2}

≤ φ(K)e

−|x|2 /4T

3T /4

√ 0

du ≤ φ(K)T p2T (x). u(T − u)

Altogether we get ds τ (s)(|ps1 +σ(s) − ps4 | ∗ gT −s4 )(x) ≤ φ(K)T p2T (x). {s1 +σ<s4 /2}

It remains to estimate the integral over {s1 + σ(s) ≥ s4 /2}. In that case it would be disastrous to take the absolute value inside. Instead we use the elementary estimate |pu (x) − pv (x)| ≤ c|u − v|v −1 p2v (x), when v/2 ≤ u ≤ v. Using this, we get ds τ (s)(|ps1 +σ(s) − ps4 | ∗ gT −s4 )(x) {s1 +σ≥s4 /2}



1 ds τ (s)(s4 − s1 − σ) p2s4 ∗ gT −s4 (x) s4 1 ≤ φ(K)p2T (x) ds τ (s)(s4 − s1 ) = φ(K)T p2T (x), s4

≤

the integrals of course all restricted to 0 ≤ s1 ≤ s2 ≤ s3 ≤ s4 ≤ T . Altogether, we have I2 ≤ φ(K)T p2T (x).

(1.24)

The estimates (1.23), (1.24) and (1.20) prove Lemma 1.6. In order to prove Proposition 1.4, there are still the other contributions in (1.18) to estimate, but we will not give the details here, and refer to [9]. There, the following inequalities are proved:

28

1 On the construction of the three-dimensional polymer measure ∆s1

0

ds

s1

∆s2

0≤s1 ≤s2 ≤T

x ×

T − κ1 (ε)

ds

0

s

0

T −s

x

0

≤

√

φ(K)

T

0

2T

x

The estimates for this are somewhat easier than for the ,,three leg” divergency, because the loop diagram splits off the rest immediately, and one has only to take care of the adjustments of the times. On the other hand, the divergency is more serious than in second summand of (1.18). In the course to prove the above estimate one needs K at one place and not just K0 . There then remains to estimate the last diagram, where counterterms are appearing:

∆s1

ds

0

s1

∆s4

∆s2

∆s3

∆s5

∆s6

x ≤ φ(K) T 3/2 p2T (x)

The crucial observation is that the corresponding diagram with free propagators is convergent, which is a somewhat tedious but elementary calculation, and then one estimates the above diagram by replacing successively the g s by the p s, and one catches only additional factors φ(K). Combining Lemma 1.6 with the above estimates, one therefore gets K0 (ε, β) ≤ βφ(K (ε, β)).

(1.25)

It then still remains to estimate  1     (pT (0) − gT (0))dT  ≤ βφ(K).  

(1.26)

0

As there is a slight slip in the argument in [9], we give the proof here. We use the same expansion which underlies the estimate (1.18). As the second and third summand on the right hand side of (1.18) are at most β 2 φ(K)T 3/4 p2T (x), and T 3/4 p2T (0) is integrable at 0, it suffices to prove

1.2 The skeleton inequalities and boundedness properties

   



1

dT

0

−



1

ds

ps1 ∗ g∆s2 (0)g∆s1 (0)

0≤s1 ≤s2 ≤T T



ds ps ∗ g∆s (0)

dT 0

29

0

ε

T

  ps (0)ds ≤ φ(K).

The left hand side of this expression is       1     du dv dt(p ∗ g )(0)g (0) − du dt(p ∗ g )(0) dvp (0) u t v u t v   ε   u+t≤1 u+v+t≤1  v≥ε   1−u−t     ≤  du dt(pu ∗ gt )(0) dv(pv − gv )(0) ε   u+t≤1−ε     1    + dudt(pu ∗ gv )(0) pv (0)dv    1−u−t u+t≤1   ≤K du dt(pu ∗ gt )(0)  u+t≤1  1  du dt(pu ∗ gt )(0) dv|pv (0) − gv (0)| +  1−u−t u+t≤1 dudt(pu ∗ gt )(0)(1 − u − t)−1/2 . +C u+t≤1

We now estimate gt by pt + φ(K)p2t ≤ φ(K)p2t and get that the above expression is     du dt(u + 2t)−3/2 (1 − u − t)−1/2 = φ(K). ≤ φ(K)   u+t≤1

(1.26) is therefore proved which implies now Proposition 1.4. We already have seen that this implies Proposition 1.3. We now show that the relevant information contained in Proposition 1.3 can be boosted by a rescaling argument due to Albeverio and Zhou [3] to cover any β > 0.

30

1 On the construction of the three-dimensional polymer measure

Proposition 1.8. Let β ≥ 0, T ≤ 1. Then there exist constants c1 (β), . . . , c4 (β) > 0 with ε (x) ≤ c1 (β)pc2 (β)T (x), ∀ε > 0. a) gT,β ε b) c3 (β) ≤ gT,β 1 ≤ c4 (β) . The basis is the following simple rescaling property Lemma 1.9. For all T, β, ε > 0 √ 2ε √ ( 2x) exp(a(ε, β, T )), gTε /2,β (x) = gT,β/ 2 where

sup |a(ε, β, T )| < ∞ for all β > 0.

ε>0,T ≤1

Proof.

   βT β2T ε κ1 (ε)− κ2 (ε) δ(ωT /2 −x) = E exp −βJ0,T /2 (ω) + 2 2  



2 √ β β 2ε βT κ1 (ε) 3/2 T κ2 (ε) δ(ωT − 2x) , = 2 E exp −√ J0,T (ω) + √ √ − √ 2 2 2 2

gTε /2,β (x)

1 2 1 √ κ1 (ε) = √ 2 2 (2π)3/2

1 √ −1 ε



1 = κ1 (2ε) + 1 − √ 2



2 , (2π)3/2

κ2 (ε) = κ2 (2ε) + 0(1). Implementing this, we get the conclusion. Proof of Proposition 1.8 a). From Proposition 1.3 we already know that there exists β0 > 0 such that the statement is true for β ≤ β0 . We now √ prove that ˆ ˆ then it is correct for β ≤ 2β. if the statement is correct for β ≤ β, √ ˆ T ≤ 1, ε > 0. Then Let β ≤ 2β,  ε,β ε gT,β (x) = E exp[−R0,T ]δ(ωT − x)  ε,β ε,β ≤ E exp[−R0,T /2 − RT /2,T ]δ(ωT − x) = (gTε /2,β ∗ gTε /2,β )(x) √ √ 2ε √ 2ε √ ≤ e2a(ε,β,T ) (gT,β/ ( 2·) ∗ gT,β/ ( 2·))(x) 2 2 √ ≤ 23/2 ea(ε,β,T ) c1 p2c2 T ( 2x), √ √ where ci = ci (β/ 2). As pa ( 2x) ≤ Cpa (x), this proves the claim.

1.2 The skeleton inequalities and boundedness properties

31

Proof of Proposition 1.8 b). The upper bound follows from part a), so it remains to prove the lower bound. First remark that from Lemma 1.9 we get 2ε √ gTε /2,β 1 ≥ c(β)gT,β/  . 2 1

(1.27)

We again use “induction” on β. Assume that the √ lower bound in Proposiˆ and assume β ≤ 2β. ˆ tion 1.8 b) is correct for β ≤ β, Let P(2) (with corresponding expectation E(2) ) be the product measure of two independent Brownian motions of length T /2. If ω1 , ω2 are two paths, we write T /2 T /2 J ε (ω1 , ω2 ) = ds dt1s+t≥ε δ(ω1,s − ω2,t ). 0

Then

0

! " ε,β ε,β ε ε gT,β 1 = E (2) exp −R0,T (ω ) − R (ω ) − βJ (ω , ω ) . 1 2 1 2 /2 0,T /2

Let Pˆ ε,β be the polymer measure (with gap ε) on paths of length T /2, and ε,β Pˆ(2) be the corresponding product measure. Then ε ˆ ε,β exp(−βJ ε ) gT,β 1 = gTε /2,β 21 E (2) 2ε √ 2 ˆ ε,β J ε ), ≥ c(β)2 gT,β/  exp(−β E (2) 2 1 2ε √  ≥ c1 (β). In order by (1.27). By the induction assumption, we have gT,β/ 2 1 to prove the result, we only have to estimate 1 ε,β ε ˆ E(2) J = ε ds dt E(2) s+t≥ε gT /2,β 21 0≤s,t≤T /2 # $ −Rε,β (ω ) −Rε,β (ω ) · e 0,T /2 1 e 0,T /2 2 δ(ω1,s − ω2,t ) 2 from above, and we again estimate gTε,β /2 1 from below with (1.27) and the induction assumption. $ # ε,β ε,β E(2) exp(−R0,T /2 (ω1 ) − R0,T /2 (ω2 ))δ(ω1,s − ω2,t ) ε ε ≤ dxgs,β (x)gt,β (x)gTε /2−s,β 1 gTε /2−t,β 1 ≤ c(β) dxpc1 (β)s (x)pc1 (β)t (x)

≤ c(β)pc1 (t)(s+t) (0) = c2 (β)(s + t)−3/2 . Integrating over s, t gives the desired claim.

32

1 On the construction of the three-dimensional polymer measure

We can already derive an important conclusion Proposition 1.10. For all β > 0 the family of measures ε }ε>0 {Pˆ1,β

is tight. Proof. The counterterms play of course no rˆ ole for the measures. So ε,β ε ε (dω) = exp(−R0,1 (ω))P1 (dω)/g1,β 1 . Pˆ1,β

Therefore for 0 ≤ t < t + s ≤ 1 by Proposition 1.8 ε |ωt − ωt+s |4 Pˆ1,β (dω) β,ε ≤ C(β) |ωt − ωt+s |4 exp(−R0,1 (ω))P1 (dω) β,ε β,ε β,ε ≤ C(β) |ωt − ωt+s |4 exp(−R0,t − Rt,t+s − Rt+1,1 )P1 (dω) = C(β) dx dy dz gt (x)gs (y − x)g1−t−s (z − y)|x − y|4 ≤ C(β)|t − s|2 . The tightness follows now by standard criteria. ε The above proposition of course implies that the measures Pˆ1,β have convergent subsequences as ε → 0. In the next section, we prove that the ε exists. limε→0 Pˆ1,β

1.3 The convergence of PTε,β , ε → 0 It is evident that the inequalities presented in Section 1.2 are not able to prove convergence. The reason simply is that the difference of the upper and lower bounds deviate by the contribution ∆s1

ds

0

s1

∆s2

∆s4 ∆s3

∆s5

∆s6

x

ε 1.3 The convergence of PT,β ,ε → 0

33

which does not go to 0 as ε → 0, but only stays finite. We would be much better off, if one of the integrations involved would be only over an interval which becomes small with ε. The idea to achieve something of this type is to differentiate with respect to the gap width ε > 0. To do this, the gap regularization is evidently much better suited than e.g. a lattice regularization. As mentioned in the introduction to this chapter, it can also be proved that the lattice regularization measure converges to the same limit. However, there are considerable additional difficulties popping up and we will not go into that. It should also be remarked that the inequalities we will get by differentiating with respect to ε > 0 are somewhat more delicate to handle for reasons which will become clear. We will heavily rely on the boundedness (and tightness) properties already obtained in order to estimate these diagrams. Let ψ : [0, 1] × R3 → R be bounded and smooth, and for 0 ≤ s < t ≤ 1 define Ψs,t : Ω → R by  t  Ψs,t (ω) = exp ψ(u, ωu )du . s

The functions Ψ = Ψ0,1 : Ω → R will be convenient for us. They form a convergence determining class, i.e. if we prove that ε lim Ψ dPˆT,β (1.28) ε→0

exists (for suitable T, β), then we have proved convergence of the measures, given of course the tightness which is already proved. We fix T = 1. Given the estimates in Proposition 1.8, we prove the convergence of the expression (1.28): Let ψ (ε) =

β,ε Ψ exp(−R0,1 )dP1 .

Proposition 1.11. For any bounded function ψ and all β > 0 there exists an integrable function i : (0, ∞) → (0, ∞) such that for any ε2 > ε1 ε2 ψ (ε2 ) − ψ (ε1 ) ≥ − i(ε)dε. ε1

The bound together with the bounds in Proposition 1.8 immediately prove Theorem 1.1 (for d = 3). Indeed as the ψ (ε) stay bounded by Proposition 1.8, Proposition 1.11 implies that lim ψ (ε) exists. This together with the tightε→0

ness proved in Proposition 1.10 proves the convergence of the measures. We fix now ψ bounded and smooth (with bounded derivatives of all desired order, say) and we write just (ε). First, we simply write ε2 d dε. (ε2 ) − (ε1 ) = ε1 dε

34

1 On the construction of the three-dimensional polymer measure

We actually do not want to prove that d

dε exists. This can be circumvented in the same way as in Section 1.2: We replace all δ function by p a s, derive the necessary inequalities and finally let go a → 0 in the end. This is evidently somewhat cumbersome to write down, so we pretend that we can work directly with the δ function. Differentiating gives d (ε) = dε



1−ε

β,ε

dsE(e−R0,1 δ(ωs − ωs+ε )Ψ ) + (βκ 1 (ε) − β 2 κ 2 (ε))(ε) (1.29)

0

β,ε The crucial point is now as follows. R0,1 of course still contains all the interactions, and we somehow want to expand that out like in the previous section. Especially, we want to expand out the interactions between the interval [s, s + ε] and its complement. This will lead to contributions which cancel the nonintegrability of the counterterms. The delicacy is coming from the fact that we are not allowed to expand the interaction of the time before s and after s + ε out in any way. Although these contributions are finite, they would, if expanded by Taylor only once or twice, lead to a destruction of all the cancellations. It is therefore better not to expand the interaction crossing the ,,loop interval” [s, s + ε]. We therefore have to control the necessary cancellations in the presence of the interactions of the time before s and after s + ε. Let β,ε ε ε ˜ s,β,ε + βJ0,s;s,s+ε =R + βJs,s+ε;s+ε,1 , (1.30) R0,1 0,1

where ˜ s,β,ε = Rβ,ε + Rβ,ε + βJ0,s;s+ε,1 − βεκ1 (ε) + βεκ2 (ε). R 0,1 0,s s+ε,1

(1.31)

As remarked above, the presence of the J0,s;s+ε,1 -summand in (1.31) is making a lot of trouble. Of course, we would like to argue that the term obtained when dropping the two last summands on the r.h.s. of (1.30) is cancelling with κ 1 (ε), and expanding these contributions once is cancelling with κ 2 (ε). However, this will be a cancellation of divergent terms (as ε → 0) and as in Section 1.2, some surgery will be needed to operate the divergency out. The crucial point is that we do not want to expand out any interaction unless it is an interaction connecting an ε-piece to something else. In this way we get estimates which after the cancellation of the divergencies become controllable for ε → 0. We give some details for the first part where the contribution coming ˜ cancels with κ . We then give a sketch how the rest is done the from R 1 cancellations with κ 2 are done. We use i as a generic function (0, ∞) → (0, ∞) which is integrable near 0, not necessarily the same at different occurrences.

ε 1.3 The convergence of PT,β ,ε → 0

35

Proposition 1.12. 1−ε ˜s dsE(e−R0,1 δ(ωs − ωs+ε )Ψ ) + κ 1 (ε)(ε) ≥ −i(ε). 0

˜ (We usually drop β, ε s at places where they obviously have to be, e.g. in R.) Proof. We set

Ysε

Ys =

δ(ωu − ωv )dudv.

= u≤s≤v≤1−ε

v−u≤ε

Then E(e−R0,1 δ(ωs − ωs+ε )Ψ ) = E(e−R0,1 δ(ωs − ωs+ε )Ψ0,s Ψs+ε,1 (1 + O(ε))). ˜s

˜s

The 1+O(ε) is just the Ψs,s+ε . There is evidently no interaction inside [s, s+ε] ˜ we have left out the interaction of the “loop” because of the gap, and in R with the rest. We take separately the expectation over δ(ωs − ωs+ε ) which is just pε (0) = −κ 1 (ε), and “glue” the second half of the path to the first, but then the interactions do no longer quite fit, because we no longer have any gap condition between the path before and after s, after having cut the loop out. To restore this, we have to correct by Ys : Using (1.31) we get E(e−R0,1 δ(ωs − ωs+ε )Ψ ) ˜s

= pε (0)E(e−R0,1−ε −βYs Ψ eβεκ1 (ε) e−β −R0,1−ε −βYs

= pε (0)E(e

2

εκ2 (ε)

(1 + O(ε))

Ψ (1 + βεκ1 (ε))(1 + O(ε| log ε|)).

There is, of course, also an adjustment of Ψ by the cutting out of the loop interval, but this give only a contribution 1 + O (ε). It is evident from the s considerations in Section 1.2 that E exp(−R0,1−ε − βYsε ) stays bounded (as ε → 0), and so we can neglect the 0(ε| log ε|) contribution as pε (0)ε| log ε| is integrable at 0. However pε (0)εκ1 (ε) is not integrable, a fact with which we are pleased as it will cancel the contribution coming from Ys . As Ys ≥ 0, we get e−βYs ≥ 1 − βYs , and therefore 1−ε ˜s ds E(e−R0,1 δ(ωs − ωs+ε )Ψ ) 0



≥ pε (0)(1 + βεκ1 ((ε)) 0

1−ε

% & ds E(e−R0,1−ε Ψ ) − βE(Ys e−R0,1−ε Ψ ) − i(ε).

36

1 On the construction of the three-dimensional polymer measure

It looks obvious that E(e−R0,1−ε Ψ ) = E(e−R0,1 Ψ ) + O(ε) = (ε) + O(ε), but I don’t know how to prove this. We would need something like a bound for d E(e−R0,v Ψ ). dv That is close to what we have done in Section 1.2, but there the integration over v was important. (That we differentiate here with respect to the upper boundary in contrast to the lower is of course of no relevance.) However, one can squeeze out of the arguments in Section 1.2 a slightly worse bound which is good enough for our purpose: Lemma 1.13. There exists δ > 0 such that |E(e−R0,1 Ψ ) − E(e−R0,1−ε Ψ )| ≤ Cε1/2+δ . We will not give a proof here as it is essentially a repetition of some of the steps of Section 1.2 (see p. 96 of [9]). End of proof of Proposition 1.12. With the help of Lemma 1.13, the proof is now easily finished: We have E(Ys e−R0,1−ε Ψ ) = dudv E(δ(ωu − ωv )eR0,1−ε Ψ ) u≤s≤v

v−u≤ε



≤

pv−u (0)E(e−R0,1−ε−(v−u) Ψ ) + 0(ε3/2 )

u≤s≤v

v−u≤ε

where we have just dropped the interactions between the interval [u, v] and the rest, and the “less than ε” interaction after readjusting time. This increases the expression. The readjustment of Ψ gives only  O (ε) dv du pv−u (0) = O ε3/2 . 0≤u≤s≤v v−u≤ε

which we can incorporate into i(ε). By Lemma 1.13, we can replace R0,1−ε−(v−u) by R0,1 , making an error which again can be incorporated into i(ε). Therefore, we get

ε 1.3 The convergence of PT,β ,ε → 0

0

1−ε

ds E(e−R0,1 δ(ωs − ωs−ε )Ψ ) 

37

˜s

 ≥ pε (0)(1 + βεκ1 (ε))(ε)  1 − β





1−ε

ds 0

  pv−u (0)dudv   − i(ε)

u≤s≤v

v−u≤ε

= pε (0)(ε) − i(ε), as εκ1 (ε) =

 1−ε 0

ds



pv−u (0)du dv + O(ε), and as pε (0) = −κ 1 (ε), this

u≤s≤v

v−u≤ε

proves Proposition 1.12. From Proposition 1.12, we get 1−ε  d ˜s 2 ds E e−R0,1 δ(ωs − ωs+ε ) [J0,s;s,s+ε + Js,s+ε;s+ε,1 ] Ψ ρ(ε) ≥ −β dε 0 (1.32) √ d × (1 + O(1/ ε)) − λ2 κ2 (ε) − i(ε). dε In order to finish the proof of Proposition 1.11, it therefore only remains to show that there is some cancellation between the first and the second summand on the right hand side of the above inequality, which leads to d κ2 (ε) is of order 1/ε, so it is clear that not something integrable in ε. dε much cancellation is needed. This helps very much, and allows for application of relatively crude estimates. On the other hand, it is also clear that the cancellation is here somewhat more subtle than the one in Proposition 1.12, because the three leg diagram is more delicate to handle than the loop one. I will not give the details here of the estimates, as it is a bit repetitive of what had been done in Proposition 1.11 (and in the last Section). Here a short outline: One of the problems is of course that J0,s;s,s+ε and Js,s+ε;s+ε,1 contain interactions which go outside the interval [s, s + ε], so they come into ˜ s . As remarked at the beginning of this section, it is not conflict with R 0,1 ˜ s which ties the part possible to cancel or expand the interaction inside R 0,1 before and that after s. However, it turns out that we can essentially neglect ˜ s which come into conflict with the above J-terms. the interactions inside R 0,1 What helps here a lot is the fact that the divergency is only logarithmic, and one can work with somewhat crude estimates. What one does is to choose some parameter 0 < γ < 1, whose value is not very important, and cut out ˜ s all the interactions with the intervals [s−εγ , s] and [s+ε, s+ε+εγ ]. from R 0,1 However, we retain (this is crucial) the interaction between [0, s − εγ ] and [s + ε + εγ , 1]. This surgery cost an error which can be incorporated into i(ε). This is essentially an argument like the one involving the Lemma 1.13 above.

38

1 On the construction of the three-dimensional polymer measure

Likewise, we drop inside J0,s;s,s+ε the interaction between [0, s−εγ ] and [s, s+ ε], and similarly for Js,s+ε;s+ε,1 . In this way, we keep the interactions inside (1.32) separated, and we can now operate the divergency out, cancelling with the derivative of κ2 . There arise now the same problems we had encountered in the last section, namely, that in contrast with the situation with the loop diagram, one has to “glue” the two loose ends together, after taking out the three-leg diagram, but this can essentially be handled in the same way as we did it in details in the last section. One then still has to restore the interaction with the now “void” interval [s − εγ , s + ε + εγ ], and one has to show that this gives again an error which can be incorporated into i(ε). The whole procedure is a bit messy and needs some care, but it should be fairly evident that with the tricks already developed, this can be done, and (1.32) can be proved in this way, leading then to Proposition 1.11. For further details, see the [9]. It should be emphasized that the above considerations do not depend on having β > 0 small. The argument are valid as long as the estimates of Section 1.2 are true, that is, according to Proposition 1.8, for all β > 0. So Proposition 1.11 follows for all β > 0. I would like to finish this chapter with two Remarks: • A shortcoming of the above argument is that we have used at various places that Ψ is a smooth function. In fact, I don’t have a proof that ε (x) exists, although there is no reasonable doubt that it is true. limε→0 gT,β To prove this would need refinements of the arguments at several places, and would probably be quite delicate. • Finally some comments about the direct x-space method developed here: Various considerations would become, of course, simpler in Fourier-space, and applying Laplace transforms in time. It is, however, somewhat delicate to implement the monotonicity properties coming from the positivity of the interaction into properties of the Fourier-transforms. Also the diagrams are most easily estimated if one has good pointwise estimates of the two-point functions. Also taking Laplace transforms in time leads in the end to the problem to invert these in order to get results for fixed time. This inversion is notoriously difficult. So, in the end, I believe that working directly in x-space and at fixed time is giving better results. Another example where direct x-space methods have been developed is the recent new method to prove diffusivity for weakly self-avoiding random walks for d ≥ 5 [18] which leads to somewhat stronger results obtained in a more direct and transparent way in comparison to the treatments developed originally (see [59]).

2. Self-attracting random walks

2.1 Introduction We discuss in this chapter a number of problems of random walks with selfattracting path interactions which are all closely related to large deviation theory. A simple case of an attraction would be just to change sign in the (weakly) self repellent case of Chapter 1. For technical reasons, it is convenient to work with continuous time but discrete state space Markov processes. Therefore, we consider the standard symmetric random walk on Zd starting in 0 having holding times with expectation 1/d. The path measure on the space D∞ = D([0, ∞), Zd ) of right continuous piecewise constant paths is denoted by P. We will also write DT for the set of paths of length T. As usual, we write Xt (ω) = ωt , ω ∈ D∞ for the evaluation map. We then transform the path measure in the same way as in the weakly repellent case, just having the opposite sign of the coupling constant: 

 T T def PˆT,β (dω) = exp β ds dt1ω =ω P (dω) ZT,β , β > 0, t

0

s

0

However, it is easy to see that this is not an interesting object, as the selfattraction is far too strong. In fact,  a path staying just all the time at 0 up to time T gets a weight exp βT 2 , whereas the entropic cost for doing that is only of order exp [−cT ] . It is therefore evident that as T → ∞ the path measures just concentrates with probability going to 1 on the path identical to 0. A more interesting object is obtained when having the interaction only of order 1/T. Therefore, we define for β > 0: 

 T β T def ˆ PT,β (dω) = exp ds dt 1ωt =ωs P (dω) ZT,β . (2.1) T 0 0 This path measure has been investigated in two papers [24] and [14]. In the first one, it was shown that the for d ≥ 2, the measure behaves diffusively if β is small enough (actually for discrete time walks), and in the second, it was shown that for d = 1, and in all dimensions if β is large enough, the path measures is localized in the sense that the end points ωT have fluctuations of order one, but these fluctuations stay non-trivial in the T → ∞ limit.

40

2 Self-attracting random walks

Therefore, for d ≥ 2, there is what is called a collapse transition if β grows from small values to large ones. We will give the argument for the diffusive behavior in Section 2, and discuss the localized phase in Section 3. There are other models which have a similar behavior. One case is Brownian motion transformed by the Wiener sausage in such a way that large volumes of the sausage are suppressed. For a random walk the rˆ ole of volume of the sausage is played by the number NT (ω) of sites visited up to time T , and for these, this would correspond in transforming the path measure P in the following way: def

dPˆT,β (ω) = exp [−βNT (ω)] dP (ω)/ ZT,β , where

(2.2)

def

ZT,β = E (exp [−βNT (ω)]) . It had been proved in [11] (and in [72] for the Wiener sausage) that for d = 2 the path measure is concentrated on paths which stay inside a ball of radius of order T 1/4 . This is closely related to the classical analysis of Donsker and Varadhan of the leading order asymptotic behavior of ZT,β . Sznitmans results and techniques have been extended recently to arbitrary dimension by Povel [62]. We will give a discussion of these results in Section 2.5. Sznitman’s approach uses the enlargement of obstacles techniques (see [73]). The approach in [11] is more combinatorial by “path counting”, and is rather involved. The problem amounts to a droplet construction, where the macroscopic droplet is trivial, namely just a ball. It is remarkable that one can prove that in all dimensions the microscopic droplet approaches the macroscopic one in L∞ −norm (at least from “outside”), whereas the corresponding analytic variational problem is stable only in L1 (for d ≥ 3). We will not be able to present the details here, but we will give a discussion of this aspect. This model has no collapse transition: For all β > 0, the path measures lives on a droplet of scale T 1/(d+2) . However, an interesting and somewhat unexpected features shows up if we make the self-attraction weaker by replacing β by a coupling constant which goes to 0 as T → ∞. Fix α > 0 and define

  β def ZT,β,α = E exp − α NT (ω) . T One way to estimate this is just by Jensen’s inequality, which gives the trivial estimate   β ZT,β,α ≥ exp − α E (NT (ω)) . T It is well know that for d ≥ 3, asymptotically E (NT (ω)) ∼ dκT, where κ is the escape probability for a discrete time random walk from a single point.

2.1 Introduction

41

(The factor d is coming from the holding times having expectation 1/d). For α = 0, this estimate is very bad as it is known from the  classical work of  Donsker and Varadhan [34] that ZT ≈ exp −cT d/(d+2) . It turns out that the Jensen estimate is essentially sharp as soon as α > 2/d. Similar to (2.2), we can define a path measure   . def P-T,β,α (dω) = exp −βT −α NT (ω) P (dω) Z, but these measures have not (yet) been investigated in the literature. The fact that the Jensen estimate is essentially sharp for α > 2/d suggests that this path measure is just diffusive in this regime. As α crosses 2/d, there is a collapse transition: Jensen’s inequality is no longer sharp and in fact ! "    log E exp −βT −α NT (ω) ∼ exp −const × T (d−2α)/(2+d) , (2.3) log

∼ meaning that the quotient of the logarithms tends to 1. This is been proved in [10] and [70]. The somewhat strange exponent of T will become clear in Section 2.5. The Povel result suggests, but this has not been proved, 1+α that for α < 2/d, the path measure is localized on scale T 2+d , and for α > 2/d, it is just diffusive. Remark that the critical case α = 2/d (d ≥ 3), would correspond to the path measure living on a subdiffusive scale T 1/d . This critical case has recently been investigated in [6] and there are some quite interesting features. For instance, it turns out that there is a collapse transition from small to large β. I will discuss this critical case in Section 2.6, but again, up to now, the path measures have not been investigated, but only the “free energy”, i.e. an asymptotic evaluation of the type (2.3). There are several motivations for the investigation of these problems. In the physical literature, the main interest in collapse transitions are for models which have a mixed attractive and repulsive interaction. Mathematically, essentially nothing is known, not even about the diffusive behavior in high dimensions. For the physical background, see [20], [21]. As an example, consider the interactive random walk (in discrete time, say), defined by  

1 Pˆn,β,γ (ω) = exp −β 1ωi =ωj + γ 1|ωi −ωj |=1  , Zn,β,γ 1≤i<j≤n

1≤i<j≤n

β, γ > 0. One would expect that if γ  β, and at least in high dimension, the repulsion dominates the attraction, and the model would just be diffusive. There is however no proof of this, and it appears that the lace expansion with which the diffusive behavior for γ = 0 has been proved is completely powerless as soon as there is a positive γ. In the physical literature, there is a collapse transition predicted if one changes the parameter γ. Some of the investigations above had been motivated by the long-standing open problem to determine the effective mass of the so-called polaron in the

42

2 Self-attracting random walks

strong coupling regime, which is a (one-dimensional) model with a Kac-type potential and a continuous symmetry (actually a shift degeneracy). This is closely related with some of the models discussed here, but it is still (mathematically) an open problem. For a heuristic derivation, see [68]. We will give a short discussion of this in the final section of this chapter. It is not possible to give proofs of all the results presented and I will concentrate on those parts which I think are the most instructive ones. In particular, I give a proof of the probabilistic part of the evaluation of the limit of the model (2.1). The proof presented here incorporates some simplifications of the one given in the original paper [14]. To finish this introduction, I state some of the relevant facts on large deviations, which will be used during this and the next chapter. For proofs, see e.g. [43]. Let YT , T ≥ 0, be a stochastic process, where T is either ∈ N or ∈ R, which takes values in a Polish space Σ. Let further (aT )T ≥0 be an increasing, real-valued, positive function of T. One says that (YT ) satisfies an (aT )large deviation principle (LDP) with rate function I : Σ → [0, ∞], if the following properties are satisfied: L1 I is upper semi-continuous and has the property that for any a > 0, the set {x ∈ Σ : I (x) ≤ a} is compact in Σ. L2 For any closed subset A ⊂ Σ, one has lim sup a−1 T log P (YT ∈ A) ≤ − inf I (x) . T →∞

x∈A

L3 For any open subset A ⊂ Σ, one has lim inf a−1 T log P (YT ∈ A) ≥ − inf I (x) . T →∞

x∈A

There are a number of cases, where only a weaker form is valid, namely where L1 is replaced by just the semi-continuity condition, and L2 is required only for compact subsets A of Σ. In such a case one says that a weak large deviation principle holds. A consequence of a LDP is the following result, called Varadhan’s Lemma Lemma 2.14 (Varadhan). Assume that (YT ) satisfies an (aT )-LDP with rate function I. Then for any continuous function F : Σ → R which is bounded above, one has lim

T →∞

1 log E (exp (aT F (YT ))) = sup (F (x) − I (x)) . aN x∈Σ

Remark 2.15. If F is only upper semi-continuous, then one gets an upper bound for the limit superior, and if F is lower-semicontinuous, one gets a lover bound for the limit inferior.

2.1 Introduction

43

Here the examples which are important for us. The first is the SanovTheorem: Let Xt , t ∈ N, be a sequence of i.i.d. random variables, taking values in a Polish space S, with law µ. The so-called empirical process is defined by def

LT =

T 1
δX . T t=1 t

+ LT takes values in the space M+ 1 (S) with the weak topology. M1 (S) is a Polish space itself (more precisely, there is a metric, e.g. the Prohorov metric, which metricizes the weak topology, and with which the space is Polish). Then we have

Theorem 2.16 (Sanov). (LT ) satisfies a T -LDP with rate function I given by

 dν def dν, I (ν) = k (ν | µ) = log dµ where I (ν) is defined to be ∞ if ν is not absolutely continuous with respect to µ or if the logarithm of the derivative is not integrable. Another case of great importance for us is the celebrated large deviation theorem for the Brownian motion by Donsker and Varadhan. For this, we consider a d-dimensional Brownian (βt )t≥0 , and again the empirical distribution T def 1 LT = δβt dt. T 0 Theorem 2.17 (Donsker-Varadhan). (LT ) satisfies a weak T -LDP with rate function I given by 2 def ∇f 2 , I (v) = 2  where f = dν/dx. I (ν) is defined to be ∞ if ν is not absolutely continuous with respect to Lebesgue measure, or when f is not (weakly) differentiable with gradient in L2 . The fact that the empirical distribution satisfies only a weak LDP causes a lot of problems. One way out is often by a compactification procedure. The fact is that the Brownian motion on a compact manifold satisfies a full LDP. The case which is important for us will be the Brownian motion on a flat torus, i.e. just the Brownian motion which is wound up periodically. Convention: During this chapter, we again use C as a generic positive constants not necessarily the same at different occurrences. It may depend on the dimension, and on a fixed coupling constant β, but on nothing else, except when indicated clearly. In contrast, we use c1 , c2 , . . . for positive constants which stay fixed after having been introduced.

44

2 Self-attracting random walks

2.2 A maximum entropy principle To start with, we consider the following trivial problem. Let X1 , X2 , . . . be a sequence of independent coin tossings: P (Xi = 0) = P (Xi = 1) = 12 . If α > 12 , then by the Bernoulli law of large numbers n

P (Sn /n ≥ α) → 0,

as n → ∞, where Sn = i=1 Xi . Question: what is the limiting distribution of X1 , if we condition on the event {Sn /n ≥ α}? The answer is evident: lim P (X1 = 1 | Sn /n ≥ α) = α.

n→∞

Similarly, the conditional distribution of X1 , . . . , Xk(n) converges (in total variation) to coin tossing if k(n) = o(n). (This can of course not be true for k (n) = n). We consider a slightly more general problem. We assume that the Xi are i.i.d. random variables, taking values in a Polish space S equipped with its Borel field S. P is the product measure of the law µo of the Xi on Ω = (S, S)N , with the Xi being the projections Ω → S. The empirical distribution is 1
Ln = δX . n i=1 i n

Let further F : M+ 1 (S) → [−∞, ∞) be an upper semicontinuous function. We will assume that F is bounded above, but it may take the value −∞. We consider the transformed measure on Ω 1 exp [nF (Ln )] dP. dPˆn = Zn Zn = E (exp[nF (Ln )]) By Sanov’s Theorem, and the upper semicontinuity of F, we have 

  dµ 1 def dµ , (2.4) lim sup log Zn ≤ bF = sup F (µ) − log dµo n→∞ n µ and if Flc is the lower semi-continuous modification of F, then lim inf n→∞

1 log Zn ≥ bFlc . n

(2.5)

Proposition 2.18. Assume bF = bFlc > −∞ (which in particular is true if F is continuous). Then the sequence {Pˆn } is tight in the weak topology on the set of probability measures on Ω. Any limiting probability measure Q has a  representation Q = µN Γ (dµ), where Γ is a probability measure on M+ 1 (S) which is concentrated on /

 0 dµ KF = µ : F (µ) − log dµ = bF dµo

2.2 A maximum entropy principle

45

Remark 2.19. This is a very weak formulation of a so called propagation of chaos result. For much stronger variants (under more restrictive conditions on F ), see [5]. Proof. The proof is a very easy application of the Sanov Theorem. As the rate function has compact level sets, it follows that KF is a compact subset of M+ 1 (S). Moreover, if Uε (KF ) is an open neighborhood of KF , it follows from (2.4), (2.5) and the assumption bF = bFlc that lim Pˆn (Ln ∈ Uε (KF )) = 1.

(2.6)

n→∞

/ Uε (KF )) = E (exp [nG(Ln )])/ E (exp [nF (Ln )]) , where we In fact, Pˆn (Ln ∈ def

c

set G = F on (Uε (KF )) and −∞ otherwise. Then the denominator behaves in leading order as exp [nbF ] , !whereas the numerator can be estimated from  "  F (µ) − log (dµ/dµ above in leading order by exp n supµ∈U ) dµ  o / ε (KF ) exp [nbF ] .  From (2.6) it follows that the sequence Pˆn L−1 of probability mean n≥1

sures on M+ 1 (S) is tight and any limit measure is supported by KF . Now, we decompose  Pˆn (·) = Pˆn (·|Ln ) d Pˆn L−1 . n M+ 1 (S)

Evidently, we have Pˆn (·|Ln ) = Pn (·|Ln ) , which is just drawing without replacement. It is well know that for large n, drawing without replacement is asymptotically the same as drawing with replacement, if we consider only o(n) drawings (which is much more than we need for weak topology considerations). Therefore, in the weak topology (and also in some stronger ones), ˆ Pn (·|Ln ) is close to LN n . From this, we easily see that the sequence {Pn }n≥1 is tight (as a sequence of probability measures on Ω), and every limit point is of the required form. The above Proposition evidently applies to the coin tossing example at the beginning. The empirical distribution there is just the relative number of 1’s in the sequence, and we take F = 0 if this is ≥ α, and −∞ otherwise. It should however be remarked that already quite simple modification of this trivial example can become quite delicate, as is revealed by the following example (see [29]): Exercise 2.20. Start with the coin tossing sequence of length n as above, and define n−1
1{Xi =1,Xi+1 =1} . Tn = i=1

46

2 Self-attracting random walks

Then determine lim P (X1 = 1 | Tn /n ≥ α)

n→∞

for α > 1/4. The exercise falls into a category of problems running under the heading “equivalence of ensembles”, in that case between some type of microcanonical and grand canonical ones. There are still many open problems in this field (see for instance [53]). Remark 2.21. If KF contains just one point, say µ, then the Proposition states that Pˆn converges to µN . If KF contains more than one point, then one usually has to derive finer asymptotics in order to evaluate the limit law of Pˆn . The situation we encounter in some of the following sections is more delicate than the one in Proposition 2.18, mainly because there KF contains more than one point (and is not even compact). The models we discuss in this (and to some extent also in the next) chapter are all variations of the above situation: The path measure of a “simple” process is transformed by a density of the form exp [H (path)] dP/E exp [H (path)] , where the “Hamiltonian” H is given by some self-interaction of the path (or in the next chapter by some interaction with a “wall”. What large deviation theory in these examples provides, is an asymptotic evaluation of the normalizing E exp [H (path)] , usually up to logarithmic equivalence. This is usually far from sufficient to determine exactly the path measure, which will be the main task in some of the next sections. The above Proposition 2.18 is in this respect a bit misleading. Let us now start with discussing the self-attracting random walk. P is the law of the standard symmetric random walk (in continuous time), starting at 0, with holding times of expectation 1/d, and we define the transformed ˆ path measure PˆT,β by (2.1). It is formally convenient to have  PT,β defined as a measure on paths of infinite length, i.e. on D∞ = D [0, ∞), Zd . Of course, after time T it is just an ordinary random walk. Remark that the T T 2 Hamiltonian T1 0 ds 0 dt 1ωt =ωs can be written as T lT 2 where lT is the normalized local time: T def 1 1{Xs =x} ds, lT (x) = T 0  2 and lT 2 = x lT (x)2 . Clearly, lT is a random probability measure on Zd . It satisfies a weak LDP (see e.g. [30]):

2.2 A maximum entropy principle

47

 d with rate Proposition 2.22. (lT )T ≥0 satisfies a weak LDP in M+ 1 Z    1 2 function I(µ) = 2 x,y ( µ(x) − µ(y)) , where summation is over (unordered) nearest neighbor pairs x, y. From this proposition, we easily get: Proposition 2.23. " ! 1 1 2 log ZT,β = lim log E exp βT lT 2 T →∞ T T →∞ T


def 2 µ(x) − I(µ) . = b(β) = sup β lim

µ

(2.7)

x

Proof. This is essentially Varadhan’s Lemma but there is a slight problem. d If F : M+ 1 (Z ) → R is continuous, and has the property that {µ : F (µ) ≥ a} is compact for all a, then by a version of Varadhan’s Lemma we get 1 log E exp [T F (lT )] = sup (F (µ) − I(µ)) . T →∞ T µ  In our case, we take F (µ) = x µ(x)2 , but this evidently does not satisfy the above compactness property. There is however a very simple trick. Consider the periodized situation, where we replace Zd by a finite discrete torus TR = {0, . . . , R − 1}d , and correspondingly a symmetric random walk with periodic boundary conditions on this torus. We can just map the old random walk by “winding it up” in an evident way. Then we have 1 12 2 (2.8) lT 2 ≤ 1lTR 12 , lim

where lTR (x) is the local time for the wound up random walk on the torus. Now, for the random walk on the torus, we evidently have a full  LDP,because 1 R M+ 1 (TR ) itself is compact, with a rate function I (µ) = 2

x,y ( µ(x) −  2 µ(y)) , the only difference being that the summation is now over nearest neighbors on the torus. Therefore ! 1 12 " 1 1 lim sup log ZT,β ≤ lim sup log E exp βT 1lTR 12 T →∞ T T →∞ T 


def R 2 R = b (β) = sup β µ(x) − I (µ) , µ

x

R

and it is easy to see that limR→∞ b (β) = b(β). Therefore, we get lim sup T →∞

1 log ZT,β ≤ b(β). T

The lower bound is no problem and follows from the weak LDP (and the continuity of the functional).

48

2 Self-attracting random walks

It should be remarked that the above monotonicity argument is rather special. It depends crucially on (2.8). We will encounter in Section 2.4 a situation where such a procedure cannot immediately be applied, and where things become then more delicate. Having the above large deviation property, a natural first question is to ask whether or not there are minimizers of the variational problem. This is directly connected with the question if b(β) > 0. Proposition 2.24. a) If d = 1, then b(β) > 0 for all β > 0. b) If d ≥ 2 then there exists βcr (d) > 0 such that b(β) > 0 for β > βcr (d) and b(β) = 0 for β < βcr (d). Proof. Evidently, b(β) is increasing in β, and furthermore, b(β) > 0 if β is large enough. This simply follows from the fact that I(δ0 ) is finite. Therefore, it remains to prove that for d = 1, we have b(β) > 0 for all β, and that for d≥2
µ(x)2 ≤ CI(µ). (2.9) x

We start with the one dimensional case. We define a sequence of measures which become flat and flatter: µn (x) =

max(1 − |x|/n, 0)2 , ξn

where ξn is the appropriate norming. Evidently, ξn ∼ Cn. Therefore,  2 −2 . Therefore, β x µn (x)2 > I(µn ) x µn (x) ∼ C/n, and I(µn ) ∼ Cn for any β > 0 if n is large enough. This proves a). The inequality (2.9) follows from the (discrete version of the) Sobolev inequality 4 2 2 g4 ≤ C g2 ∇g2 , applied to µ(x) = g 2 (x), which holds for d ≥ 2. Here ∇ denotes the discrete gradient. It turns out that if b(β) > 0, then there exist solutions of the variational problem. Let 2 3
def + d 2 µ(x) − I(µ) = b(β) . (2.10) Kβ = µ ∈ M1 (Z ) : β x

One of the basic difficulties we will encounter is that Kβ is shift invariant: Any shift of an element of Kβ is again in Kβ . We summarize the basic facts about this set.

2.2 A maximum entropy principle

49

Proposition 2.25. Assume b(β) > 0. Then a) Kβ = ∅. b) Any µ ∈ Kβ satisfies µ(y) > 0 for all y ∈ Zd . c) There exist C > 0 such that for any µ ∈ Kβ there exists xµ ∈ Zd with µ(y − xµ ) ≤ C exp [−|y|/C] for all y. The proof is not difficult, but a bit lengthy. I will not give it here (see [14]). A natural question is if there is uniqueness modulo shifts as soon as b(β) > 0. Unfortunately, I don’t know the answer, not even for d = 1. Corresponding uniqueness questions for variational problems in the continuous setting on Rd have a long history with many results. However, the knowledge about similar questions on Zd is essentially zero. One of the difficulties in the discrete situation is that standard symmetrization techniques do not work. The discrete problems seem to be inherently more delicate than the continuous ones. Take for instance the variational problem in the one-dimensional case, but in the continuous situation. This just is the problem to maximize 1 g (x)2 dx, β g(x)4 dx − 2  subject to the condition g(x)2 dx = 1. It is easy to see that modulo shifts, there is just one solution of the Euler equation 4βg(x)3 + g (x) = λg(x) (2.11)   which decays to 0 at infinity and satisfies g(x)2 dx = 1 (just β/2/ cosh(βx) and its shifts). On the other hand, the Euler equation of the discrete problem, namely 4βg(x)3 + ∆g(x) = λg (x) (2.12)  2 2 subject to g(x) = 1 (we have replace µ(x) by g(x) ) has countably many such solutions. ∆ is the discrete Laplacian. I have no formal proof of this, but playing on the computer one “sees” them, and it is probably not difficult to prove it. (Computer simulations however indicate that among these solution there are just two candidates as maximizers. Both are symmetric, although we don’t have a proof that the maximizers have to be symmetric.). Anyway, one easily gets convinced that (2.12) is a much more delicate equation than (2.11), and even more so the discrete variational problem. It is not difficult to see that one has uniqueness if β is large enough. This is just coming from the fact that for “β = ∞”, the solutions are unique modulo shifts, namely just the δx . By a perturbation argument around β = ∞ one can prove that uniqueness persists for large β:

50

2 Self-attracting random walks

Proposition 2.26. If β ≥ 2d then & % Kβ = θx µ : x ∈ Zd + + d d d for some µ ∈ M+ 1 (Z ), where θx : M1 (Z )→M1 (Z ) is the usual shift θx µ(y) = µ(y − x).

This is Proposition 1.19 of [14]. I will not repeat the proof here. There is actually nothing special about 2d. The proof allows for slightly lower values of β. (The proof in [14] can actually be simplified somewhat, if one is satisfied with a statement “for large enough β ...”.) In view of the Proposition 2.18, it is natural to conjecture that for β > βcr (d), one has that PˆT,β behaves for large T such that lT is close to some element in Kβ . This is in fact true (see Proposition 2.35 below). What makes things delicate is that Kβ contains infinitely many elements. It will turn out that there are infinitely many elements of Kβ which will get positive limiting weight under PˆT lT−1 . In the case where one has uniqueness modulo shifts, we will actually prove that all elements of Kβ get positive weight. However, for µ ∈ Kβ which lie for out, these weight will be small, uniformly in T. A preformulation of the main result on this collapsed phase is the following: Theorem 2.27. Assume b(β) > 0. Then  + d a) PˆT,β lT−1 is tight in M+ 1 (M1 (Z )). T >0

b) There exists c(β) > 0 such that sup exp [c(β) ωT ] dPˆT,β < ∞. T

c) If there is uniqueness modulo shift, then lim PˆT,β lT−1 , lim PˆT,β ωT−1

T →∞

T →∞

and lim PˆT,β T →∞

exist. The exact formulation of the limits needs a bit of preparation, and we will give it later. limT →∞ PˆT,β is understood in the sense of weak convergence on the path space. In particular, it does not imply the existence of limT →∞ PˆT,β ωT−1 which has to be treated separately. The latter is the more interesting object. The existence of the limit means that there is no rescaling for ωT , so that this random variable stays stochastically bounded under PˆT,β as T → ∞. It is natural to conjecture that for β < βcr , PˆT,β just behaves diffusively, but there is no full proof of that. What Brydges and Slade in [24] proved is that for d ≥ 2 there exists βo (d) ≤ βcr (d) such that for β < βo (d) there is diffusive behavior (with some complications for d = 2). It could actually well be that the βo (d) they define is exactly βcr (d), but I don’t know how to prove this. We will threat the diffusive behavior in the next section. In Section 2.4, we will then come to the large β i.e. collapsed case.

2.3 The diffusive phase for self-attracting random walks

51

2.3 The diffusive phase for self-attracting random walks I am presenting part of the arguments in [24] for the existence of a diffusive phase for dimensions d ≥ 2. The two dimensional case is the most interesting one, and it is related to the topics discussed in Chapter 1. I give a detailed discussion of the case d ≥ 3, and will add some comments about the twodimensional case. For abbreviation, we set γT =

1 T





T

T

dt 0

0

2

ds1ωs =ωt = T lT 2 .

Let ps (y) be the transition probabilities for our random walk. Then one has the estimates " ! ps (y) ≤ C min 1, |s|−d/2 exp [−|y|/Cs] (2.13) and def



∞

G(y) =

 ps (y)ds ≤ C min |y|−d+2 , 1

(2.14)

0

for d ≥ 3. ((2.13) is actually very crude, but it suffices for our purpose). Lemma 2.28. Assume d ≥ 3. a) There exists βo (d) > 0 such that sup E (exp [βγT ]) < ∞

T >0

for β < βo . 2 b) E (γT − EγT ) = o(1) as T → ∞. Proof. a) By Jensen’s inequality, we have exp [βγT ]

 n+1 ∞ T 1

βn lT (y) exp [βT lT (y)] = ds1ωs =y ≤ T y n=0 n! 0 y ∞ 1

n = β (n + 1) 1{ωs =y,...,ωs =y} ds1 . . . dsn+1 . 1 n+1 T y n=0 0≤s1 <...<sn+1 ≤T


Therefore,

52

2 Self-attracting random walks

E (exp [βγT ]) ∞ 1

n ≤ β (n + 1) ps1 (y)ps2 −s1 (0) . . . psn+1 −sn (0) T y n=0 0≤s1 <...<sn+1 ≤T ∞ 1
n β (n + 1) ps2 −s1 (0) . . . psn+1 −sn (0) = T n=0 0≤s1 <...<sn+1 ≤T ≤

∞


β n (n + 1)G(0)n ,

n=0

which is finite if β < G(0)−1 . This proves a) In order to prove b), remark first that T 2
T ds dt ps (y)pt−s (0) T y 0 s T 2 T = ds dt pt−s (0) = 2G(0) + o(1). T 0 s

EγT =

Therefore, we have to prove  E γT2 = 4G(0)2 + o(1). 4 γT2 = 2 ds1 dt1 ds2 dt2 1ωs1 =ωt1 1ωs2 =ωt2 . T 0≤s1
2.3 The diffusive phase for self-attracting random walks

53

 For d ≥ 4, we can estimate the r.h.s. by (1/T ) y G(y)3 = O(1/T ), but for d = 3, this is divergent, and one has to argue slightly more carefully. Using (2.13), one gets 3  

 T 1 C
C
|y| ps (y)ds ≤ min , exp − R2 (T ) ≤ T y T |y|3 CT 0 y =0

 log T =O . T This proves the Lemma. With this Lemma, one now easily gets the following result: Theorem 2.29. Assume d ≥ 3 and β < βo (d). Then, using Brownian scaling, √  def ρT (ω)(t) = ω(tT )/ T , ω ∈ D [0, ∞), Rd , one has

lim PˆT,β ρ−1 T = P∞ ,

T →∞

weakly, where P∞ is the standard Wiener measure. Proof. This is immediate from the estimates in the Lemma 2.28: Let Φ :  D [0, ∞), Rd → R be continuous and bounded. Then   exp [βγT ] (Φ ◦ ρT ) dP  lim Φ d PˆT,β ρ−1 = lim T T →∞ T →∞ exp [βγT ] dP  exp [β(γT − E(γT ))] (Φ ◦ ρT ) dP  = lim T →∞ exp [β(γT − E(γT ))] dP = lim (Φ ◦ ρT ) dP = ΦdP∞ . T →∞

The third equality is coming from the fact that γT − ET (γT ) converges to 0 in probability according the Lemma 2.28 b), and the necessary exponential moment estimate are from Lemma a). Remark 2.30. It is also not difficult to prove that one has convergence of all moments of finite dimensional distributions (see [24]). The two dimensional case is more delicate and the limiting measure is more interesting. I will give only some short comments about this case. We step back to the discussion of Chapter 1. There we had argued the for Polym = d = 2, 3, the (formal) rescaling property of the polymer measure dPˆT,β Wiener exp [−βJ0,T ] dP /Z, which is

54

2 Self-attracting random walks Polym Polym −1 PˆT,βT = Pˆ1,β , −(4−d)/2 ρT

indicates that the self-repellent random walk with coupling parameter βT −(4−d)/2 should converge, after a Brownian scaling, toward the polymer measure with coupling parameter β. This is the content of Theorem 2.29 for d = 3 and has been proved for d = 2 by Stoll [69]. The renormalization needed to define the two dimensional polymer measure is just the subtraction of the logarithmically divergent loop diagram, i.e. just by subtracting the expectation. This is an old result of Varadhan [75] who proved (with a different regularization) that  ε ε YT = lim J0,T − EJ0,T ε→0

exists in L2 , and E exp [−βYT ] < ∞ for all β ≥ 0. The polymer measure for d = 2 is therefore just Polym dPˆT,β = exp [−βYT ] dP Wiener /Z.

Somewhat surprisingly, YT has also a positive exponential moment, as has been proved by LeGall: Proposition 2.31. ([48]) There exists βo (2) > 0 such that E (exp [βY1 ]) < ∞ for β < βo (2). Therefore, the polymer measure exists (for d = 2 not for d = 3) also with the “wrong sign” if β is small. This makes it plausible that Stoll’s result stays correct also in the attractive case. This is in fact true and is the content of the following result by Brydges and Slade: Theorem 2.32. Assume d = 2. Then there exists βo (2) > 0 such that for 0 ≤ β < βo (2) a) supT >0 E RW exp [β (γT − EγT )] < ∞ ˆ Polym c) limT →∞ PˆT,β ρ−1 T = PT,−β I will not give the details. There are a number of interesting observations: • The renormalization is necessary. Evidently E RW exp [βγT ] → ∞ for β > 0, simply because γT → ∞ in probability. • In contrast to the situation for d ≥ 3, the limit in the two dimensional case depends on β.

2.4 The collapsed phase for self-attracting random walks

55

2.4 The collapsed phase for self-attracting random walks In this section, we will discuss some aspects of Theorem 2.27. We will focus on the limiting behavior of PˆT lT−1 , from which actually the other properties can be derived. Therefore, we assume d = 1 or d ≥ 2 and β > βcr (d) . To simplify matters, we will also stick to the case where the variational problem (2.2) has a unique maximizer, which makes things technically a bit simpler. In that case, we can give a more precise version of the theorem, identifying the limit, which we had not been able to do in the case where one has nonuniqueness. For the rest of this section, we therefore assume & % Condition 2.33. Kβ = θx µo : x ∈ Zd for some µo . This especially applies to β ≥ 2d, but it may well always be true. The proof given here incorporates some technical simplifications compared with the one given in [14] which make it more transparent (I hope). Theorem 2.34. Assume b (β) > 0 and Condition 2.33. Then 

 −1 ˆ µo (−x)δθ µ µo (x) a) lim PT l = T →∞

T

x

x∈Z d

√ µo ∗ b) lim PˆT XT−1 =



T →∞

µ 4o

o

d 5
x∈Z  √ µo ∗ µ 4o (x) ,

x∈Zd def

where µ 4o (x) = µo (−x) . It is also not difficult to determine limT →∞ PˆT itself which turns out to be a mixture of ergodic Markov processes. This is in spirit very close to Proposition 2.18. I will give some more comments later. Remark that the choice of µo , which is of cause arbitrary in Kβ does not influence the right-hand side of the above equations. The proof of the theorem splits into three parts, which will be presented in the subsections, but I will concentrate on the probabilistic aspects. A crucial first step is to prove that under PˆT , the local times lT concentrates with high probability to a neighborhood of Kβ . We had coined this the “tube property”, because K in the case of uniqueness and d = 1 is a sort of an infinite line, so a neighborhood looks like a tube. d On M+ 1 (Z ) we take the total variation norm ·TV , which can be taken as the metric for the weak convergence (as Zd is countable). If A is a subset d of M+ 1 (Z ), then we write Uε (A) for the ε-neighborhood in total variation of A. So the statement is Proposition 2.35. For any ε > 0 / Uε (Kβ )) = 0. lim PˆT,β (lT ∈

T →∞

56

2 Self-attracting random walks

Remark 2.36. The proof will actually show that for any positive ε, the prob/ Uε (Kβ )) decays exponentially in T. ability PˆT,β (lT ∈ Even after having proved this “tube property” the path measure could still float around very freely. The next and crucial step is the proof that this does not happen. Proposition 2.37. For any η > 0 there exists S(η) ∈ N such that for all ε>0 lim sup PˆT,β (lT ∈ / Uε ({θx µo : |x| ≤ S(η)})) ≤ η. T →∞

From this tightness property, the convergence follows as will explained in subsection 2.4.3. 2.4.1 The tube property: Proof of Proposition 2.35 It is quite evident that Proposition 2.35 should be true, but there is a very annoying problem to prove it. First observe that  ! " 2 E 1lT ∈U / exp βT lT 2  ! " Pˆ (lT ∈ / U) = . 2 E exp βT lT 2 d For the estimation of the numerator, we define F : M+ 1 (Z ) → [−∞, ∞) by 2 F (µ) = µ2 if µ ∈ / U and F (µ) = −∞ otherwise. As F is upper semicontinuous, we would expect to get

lim sup T →∞

 " ! 1 1 2 log E 1lT ∈U = lim sup log E (exp [βT F (lT )]) / exp βT lT 2 T T →∞ T ≤ sup {βF (µ) − I(µ)} $ # 2 = sup β µ2 − I(µ) µ∈U /

The right hand side of this is evidently strictly smaller than b(β), so this would prove the claim. The above inequality is however not quite evident because we only have a weak LDP at our disposal. We can try to remedy the situation by using a compactification argument, i.e. wind the random walk on the torus in the same way as we did in the proof of Proposition 2.23. The problem is that in our situation, the monotonicity argument does not work out such nicely. We would like to argue as follows: Fix some (large) R ∈ N and consider again the wound up random walk on the torus TR = d {0, . . . , R − 1} . Denote # the corresponding set of solutions of$the variational 2 R R problem by K R = µ ∈ M+ 1 (TR ) : β µ2 − I (µ) = b (β) . For a given

2.4 The collapsed phase for self-attracting random walks

57

neighborhood Uε of K we would like to find ε > 0 such that for any large enough R ! ! 1 12 " " 2 1 R1 R ∈U (2.15) 1lT ∈U / ε (K) exp β lT 2 ≤ 1lT / ε (K R ) exp β lT 2 , in which case we would get the desired inequality, by estimating ! 1 12 "   1 2 1 R1 log E 1lTR ∈U ≤ sup β µ2 − I R (µ) / ε (K R ) exp β lT 2 T →∞ T R µ∈U / ε (K ) lim

which is easily seen to be < b(β) if R is large enough. However, (2.15) is not quite correct, as evidently there are probability measures which on the full space are far away from K, but which become close when wound up. On the other hand, it should be clear that such measures must be somewhat weird, and one should be able to control the event where (2.15) fails separately. This is indeed the case. The proof is based on a reflection trick. Let i ∈ N+ , 1 ≤ m ≤ d and consider the space of paths DT (right continuous pure jump) of length T. We define a reflection ϕm,i (ω) of paths ω ∈ DT at the hyperplane Hm,i def

=

%

(i1 , . . . , im−1 , i, im+1 , . . . , id ) : (i1 , . . . , im−1 , im+1 , . . . , id ) ∈ Zd−1

&

simply by switching any excursion which moves strictly to the right of the hyperplane to the left. By the “right of the hyperplane”, we mean the halfspace of points whose m-th coordinates are > i, and accordingly for the left. Remark that we start left of the hyperplane as we assume i > 0. Therefore, after the switching, the path is at the left of the hyperplane, or on it. It is easy to estimate the density of PT ϕ−1 m,i with respect to PT . Let nT,i (ω) be the number of times, the path visits the plane, coming from outside it. Then nT ,i Lemma 2.38. dPT ϕ−1 . m,i /dPT ≤ 2

This is fairly evident, and we leave it as an exercise to prove it. The switching costs at most a factor 2 “in entropy” for every visit of the plane (see [14]). One important and easy property we are using is that “finite size” approximations of the variational problem are approximating the infinite one very well. For a proof of the following Lemma we also refer the reader to [14]. Lemma 2.39. a) limR→∞ bR (β) = b(β). b) K R is close to K in the following sense: For any ε > 0 there exists Ro (ε) such that for any R ≥ Ro one has b1) For any µ ∈ K, the wound up measure µR measure on the torus TR is within ε−distance of some ν ∈ K R .

58

2 Self-attracting random walks

b2) For any ν ∈ K R one can cut the torus open in such a way (i.e. identify it with the subset {1, . . . , R}d ⊂ Zd such that if ν is extended by 0 to the whole of Zd , it is within distance less than ε to K. The Lemma states that analytically, the tube property holds, and we have to prove the probabilistic counterpart. We first state an immediate consequence of the above Lemma 2.39 and the strong LDP on the finite torus. Lemma 2.40. Given ε > 0, there exists δ(ε) > 0 such that for all R large enough  1 lim sup log PˆT lTR ∈ / Uε (K R ) ≤ −δ(ε). T →∞ T Remark that at this stage, no uniformity in R of the estimates for PˆT lTR ∈ / Uε (K R ) is claimed. To prove such an uniformity is essentially the task we have in order to finish the proof of Proposition 2.35. The idea is as follows: Take R  1. Assume we are having a path such that lT (ω) is not close to K. We however know from Lemma 2.40 that lTR lies with large PˆT -probability close to K R . By Lemma 2.39, for large enough R, K R looks much like the translates on the torus of our basic µo ∈ K (somewhat chopped to fit it onto the torus). Therefore our path, except with very small PˆT −probability, has to distribute its lT −mass on several essentially disjoint translates of µo . The problem is of course that this may happen on an increasing number, with growing T, which looks at first glance difficult to control. Nevertheless, between these chunks of translates of µo on which lT is sitting, there must be vast regions essentially not visited. We select a hyperplane Hm,i which is not often visited. Then the reflected path has essentially the same probability as the old one (not quite, of course, but this is measured by Lemma 2.38). As we have enough “empty” space, we can choose the hyperplane in such a way that after the reflection lT (ϕm,i (ω)) is not close to K R . Therefore, such a behavior of ω is excluded by Lemma 2.40. As there are several things which have to tally, we give the details. If δ > 0 is small enough, we have by our Condition 2.33 6 Uδ (θl µo ), Uδ (K) = l∈Zd

and similarly on the torus. Of course, we cannot conclude that K R consists of shifts of one element, and for µ ∈ K, it will not be true that µR ∈ K R . However, if δ > 0 is small enough and R sufficiently large, R ≥ R0 (δ) , one has 6   U2δ θk µR Uδ K R ⊂ o , k∈TR

as follows immediately from Lemma 2.39. Therefore, if ε, δ > 0 are small enough we have

2.4 The collapsed phase for self-attracting random walks

 PˆT lTR ∈ Uε (K R ), lT ∈ / Uδ (K)   6 6 % & R R ˆ lT ∈ Uε (θk µo ) , lT ∈ / Uδ (θ µo ) ≤ PT k∈TR

≤








PˆT

lTR

∈

Uε (θk µR o ), lT

∈ /

k∈TR

6

59

(2.16)



Uδ (θ µo ) .



We claim that if ε ≤ εo (δ) (small enough) and R ≥ Ro (ε, δ) (large enough) then for any 6 Uδ (θ µo ) (2.17) µ∈ / 

with µR ∈ U2ε (θk µR o)

(2.18)

there exists a hyperplane Hm,i with 0 < i ≤ R, 1 ≤ m ≤ d such that   R 2 2 (ˆ µm,i ) ∈ / Uδ/4d (K R ), ˆ µm,i 2 − µ2  ≤ 3ε and µ(Hm,i ) ≤ 3ε, where µ ˆm,i is the measure where the mass right of the hyperplane Hm,i is reflected to the left To see this, remark that for any ε > 0, if (2.18) is satisfied, and R is large enough, then µ must, by Proposition 2.25, be concentrated up to a mass 3ε on R-periodic shifts of boxes of side length L (ε) . Of course, if just one L-box is needed, and ε is small enough compared with δ, then (2.17) cannot be satisfied. Therefore, from this property it follows that for ε ≤ εo (δ), a single L-box contains at most 1 − 9δ/10 of the µ-mass. This holds true uniformly in R (large enough). It is geometrically evident that by choosing ε > 0 small enough and then R ≥ Ro (ε, δ), we can find a hyperplane Hm,i having the following three properties • Hm,i does not intersect any of the L-boxes. • The L-boxes on the right of Hm,i when reflected to the left, do not intersect the L-boxes on the left. • The right side and the left side of the hyperplane contain at least δ/3d of the mass of µ. From these three properties, it is evident that Hm,i does the job. Let Am,i (ε) be the event that the Hamiltonian of the reflected path does not deviate more than 3ε from the unreflected, i.e.  $ # def  2 2 Am,i (ε) = lT ◦ ϕm,i 2 − lT 2  ≤ 3ε .

60

2 Self-attracting random walks

plane of reflection

L

reflected path Fig. 2.1.

Then we have 2 lTR

∈

Uε (θk µR o ), lT

∈ /

6

3 Uδ (θ µo )

(2.19)



⊂

$ 6 # R (lT ◦ ϕm,i ) ∈ / Uδ/4d (K R ), Am,i (ε) , lT (Hm,i ) ≤ 3ε , 0
implying   6 R R PˆT lT ∈ Uε (K ), lT ∈ / Uδ (θ µo ) 





$  6 # R R  ≤ Rd PˆT  (l ◦ ϕ ) ∈ / U (K ), A (ε) , l (H ) ≤ 3ε T m,i m,i T i δ/4d   0≤i≤R 1≤m≤d



(2.20) R

≤ Rd+1 max PˆT (lT ◦ ϕm,i ) ∈ / Uδ/4d (K R ), Am,i (ε) , lT (Hi ) ≤ 3ε . 0≤i≤R 1≤m≤d

We would like to replace the condition lT (Hi ) ≤ 3ε by a condition on nT,i . This can be done by still adjusting the ε. By Lemma 2.41 below it follows that for any εˆ > 0 one has for ε > 0 small enough (depending on εˆ) P (lT (Hi ) ≤ 3ε, nT,i > εˆT ) ≤ exp [−(β + 1)T ] ,

(2.21)

2.4 The collapsed phase for self-attracting random walks

61

and therefore PˆT (lT (Hi ) ≤ ε, nT,i > εˆT ) ≤ exp [−T ] . For fixed εˆ, and δ, we can choose εo (ˆ ε, δ) such that for ε < εo the above inequality is true. We can therefore replace the condition lT (Hi ) ≤ 3ε in (2.20) by nT,i ≤ εˆT, making a negligible error. Next, we estimate  R PˆT (lT ◦ ϕm,i ) ∈ / Uδ/4d (K R ), Am,i (ε) , nT,i ≤ εˆT 1  βT lT 22 R E e ; (lT ◦ ϕm,i ) ∈ / Uδ/4d (K R ), Am,i (ε) , nT,i ≤ εˆT ≤ ZT e3βεT  βT lT ◦ϕi 22 R ≤ E e ; (lT ◦ ϕm,i ) ∈ / Uδ/4d (K R ), nT,i ≤ εˆT ZT e3βεT  nT ,i βT lT 22 R ≤ E 2 e ; lT ∈ / Uδ/4d (K R ), nT,i ≤ εˆT ZT  ≤ e4βεT 2εˆT PˆT lTR ∈ / Uδ/2 (K R ) . Therefore, for given β, δ > 0 we choose εˆ small enough such that the decay  of PˆT lTR ∈ / Uδ/4d (K R ) which is guaranteed by Lemma 2.40 beats e4βεT 2εˆT , and then for ε ≤ εo (β, δ), and then R large enough, one gets the desired / Uδ (K)) , which finishes the proof of Proposition 2.35. estimate for PˆT (lT ∈ Lemma 2.41. If ζi , i ≥ 1, is a sequence of exponentially distributed random variables, with parameter 1, then for t ≤ 1   n
ζi ≤ nt ≤ exp[−nh(t)], P i=1

where lim h(t) = −∞.

t→0

Proof. This is the standard one dimensional large deviation estimate. The rate function is 

∞ 1 h(t) = sup λt − log exp[λx − x]dx = t − 1 + log . t λ≤0 0 2.4.2 Tightness: Proof of Proposition 2.37 This is the crucial step of the whole argument. It should be noted that the different places in Kβ cannot be distinguished on a logarithmic scale. In fact, it is rather evident that  ! " 2 log E exp βT lT 2 ; lT ∈ Uε (µx ) lim =1 T →∞ log ZT,β

62

2 Self-attracting random walks

for all x ∈ Zd , ε > 0. The most natural way to proceed would be an evaluation of the expression P (lT ∈ Uε (µx )) up to a factor (1 + o (1)) , and the same for ZT,β . In that case we could control the quotients. This would probably be possible but has not been done in the present context, and is probably technically delicate. We instead make use of the symmetry properties. Before we start with the formal proof, I want to explain shortly the main idea. We want to conclude that under PˆT , there is only a small probability that lT ∼ µx , x far away from 0. To do this, it suffices to get estimates for large |x| of "  ! 2  E exp βT l ˆ T 2 ; lT ∼ µx PT (lT ∼ µx ) " ! =  , 2 PˆT (lT ∼ µo ) E exp βT l  ; l ∼ µ T 2

T

o

of course only in the T → ∞ limit. As already remarked, a way to control this would be to evaluate the numerator and the denominator up to a factor (1 + o (1)) , as T → ∞. There is however a much easier way to get such a control, which in spirit resembles the Peierls argument for the Ising model. If lT ∼ µx , |x| large, then the path spends most of the time near x. It is not difficult to prove that the path has to reach the neighborhood of x relatively quickly, say after time t  T, where however t has to become large when |x| is large. We are doing now a splitting of the local time in the part before time 2 t and after that. If we calculate lT 2 , we now want to split this, too, and we use that the path before t has not much intersection with the path after t, the latter hanging around x, whereas the former doesn’t. We compare that now with the situation where the path after t would hang around 0 instead of around x. We just shift any path after time t which hangs around x by −x. Of course, we have to fit the path before t to this situation.

x

shifted path from t to T

T

t

Fig. 2.2.

2.4 The collapsed phase for self-attracting random walks

63

The crucial point is that the shifting of the main part, namely the path in [t, T ] does not cost anything, due to the shift invariance which is used heavily. Lets denote the local time after the shift by lTshift . The only thing 1 12 2 which essentially distinguishes lT 2 from 1lTshift 12 is that the former has essentially no contribution from the interaction between the path on [0, t] with the path on [t, T ] , where the latter has. It turns out that typically 1 shift 12 1lT 1 ≈ lT 2 + Ct. 2 2 Therefore, we get  ! " "  ! 2 2 E exp βT lT 2 ; lT ∼ µx
e−Ct E exp βT lT 2 ; lT ∼ µ0 , uniformly in T large. On the other hand, |x| cannot be large without t, the first time, the path reaches the neighborhood of x, is not also large. Therefore, it follows, that lT ∼ µx can happen only with small P-T probability, uniformly in T for T large. Of course, this is very hand-waiving, and we now give the details of the argument. We are proving a superficially weaker result than Proposition 2.37: Proposition 2.42. There exists εo > 0 such that for all ε ≤ εo , η > 0 there exist S(ε, η) ∈ N, To (ε, η) > 0 with   6 {lT ∈ Uε (µx )} ≤ η, PˆT  |x|≥S(ε,η) def

for T ≥ To (ε, η), where µx = θx µo . Together with the Proposition 2.35, this evidently implies the Proposition 2.37. Perhaps some comments on the rˆole of ε in our proof is in order, in particular as this point had been not so well handled in the original paper [14]. We will on several occasions switch from one ε to another. As the tube property 2.35 states that outside any of any ε-neighborhoods of Kβ there is only (exponentially) negligible PˆT -mass, we can do this freely, always after having chosen T large enough. As the T → ∞ limit will always be the first to perform, this causes no problem. def If r ∈ N, let Cr = {−r, −r + 1, . . . , r}d and for x ∈ Zd , Cr (x) = Cr + x. ∂Cr (x) is the inner boundary, i.e. ∂Cr (x) = {y ∈ Cr (x) : |yi − xi | = r for some i} . We denote by τr (x) the first hitting time of ∂Cr (x) and by ξr (x) the time the process spends on ∂Cr (x) after τr (x) before leaving it for the first time. We need some control that the process does not leave ∂Cr (x) too quickly.

64

2 Self-attracting random walks

Lemma 2.43. lim PˆT (ξr (x) ≤ ρ) = 0,

ρ→0

uniformly in T, r, x. Proof. Define

/ Yt (ω) =

ωt for t ≤ τ . ωt+ξ for t > τ

T Then {Yt }t>0 and ξr are independent under P . Let lT (x) = (1/T ) 0 1{Ys =x} . Then   ξ 2  2 (2.22) lT 2 − lT 2  ≤ C . T Therefore, "  ! "  ! 2 2 E exp βT lT 2 ; ξ ≤ ρ ≤ CβE exp βT lT 2 ; ξ ≤ ρ  ! " 2 ≤ CβρE exp βT lT 2 ; ξ ≤ 1  ! " 2 ≤ CβρE exp βT lT 2 . This proves the claim. We need a further technical Lemma: Lemma 2.44. Given η > 0, there exists ro (η) such that sup

T ≥1

∞


 √ PˆT τr (0) ≤ r ≤ η.

r=ro (η)

Proof. We introduce for t < T : def

lt,T (y) =

1 T −t



T

1Xu =y du. t

Then, as above in (2.22), we have  1 12  √  2 T lT 2 − T 1l√r,T +√r 12  ≤ C r. Therefore   √ 1 12  √ √ 2 E exp[T β lT 2 ]; τr ≤ r ≤ eC r E exp[T β 1l√r,T +√r 12 ] P τr ≤ r √  √ = eC r ZT,β P τr ≤ r . √ If the random walk reaches Cr (0) in time ≤ r, it has to make at least r jumps in this time. Applying now Lemma 2.41, the claim follows.

2.4 The collapsed phase for self-attracting random walks

65

c Given ε, let (ε) be chosen such that µo (C(ε) ) ≤ ε. The main step in the proof of Proposition 2.42 is given by the following result:

Lemma 2.45. If ε > 0 is small enough, there exists A(ε) > 0, such that for x with |x|, T, u ≥ A(ε) and ρ ∈ (0, 1]  C PˆT lT ∈ Uε (µx ), ξ(ε) (x) > ρ, τ(ε) (x) > u ≤ exp [−u/C] , ρ

(2.23)

(C may depend on β and d but on nothing else, as usual). Proof. We abbreviate ξ(ε) (x) as ξ, and τ(ε) (x) as τ during this proof. A(ε) is chosen in any case bigger than (ε). Then 0 is outside B(ε) (x). Remark first that for |x| > (ε) and lT ∈ Uε (µx ), the process can spend outside of C(ε) (x) only a total time less than a proportion of εT. Therefore, on {lT ∈ Uε (µx )} we have τ ≤ c1 εT ≤ T if ε is small enough. (We remind the reader that c1 , c2 , . . . are constants not changed after having them introduced). Therefore  2 E eβT lT 2 ; lT ∈ Uε (µx ), ξ > ρ, u < τ  2 ≤ E eβT lT 2 ; lT ∈ Uε (µx ), ξ > ρ, u < τ ≤ c1 εT. (2.24) c1 εT +ρ  2 1 dt E eβT lT 2 ; lT ∈ Uε (µx ), ξ > ρ, t − ρ < τ ≤ t ≤ ρ u  2 1 c1 εT +1 ≤ dt E eβT lT 2 ; lT ∈ Uε (µx ), Xt ∈ ∂C(ε) (x), t − 1 < τ , ρ u where in the last inequality we have used that on the set {t − ρ < τ ≤ t, ξ > ρ} we have Xt ∈ ∂C(ε) (x). We can assume that c1 εT + 1 ≤ T, so that t ≤ T in the domain of integration. We have the convex combination lT = Tt lt + TT−t lt,T , and therefore t2 (T − t)2 t(T − t) 2 2 2 lt,T 2 + 2 lt , lt,T  . T lT 2 = lt 2 + (2.25) T T T If A (ε) ≥ 1/ε, we have εT ≤ 1 for T ≥ A (ε) and therefore t ≤ CεT if t ≤ c1 εT + 1. Therefore t2 2 lt 2 ≤ CεT, (2.26) T if ε ≤ 1, which we of cause assume. We now estimate the third summand in (2.25). First observe that lt,T − lT TV ≤ t/T ≤ Cε, and therefore

66

2 Self-attracting random walks

lt,T ∈ Uc2 ε (µx )

(2.27)

if lT ∈ Uε (µx ) , and we can conclude |lt , lt,T | ≤ |lt , µx | + Cε.

 c Now, lt−1 does not charge Cl(ε) (x) if t − 1 < τ, and as µx Cl(ε) (x) ≤ ε, we conclude |lt , µx | ≤ |lt−1 , µx | + lt−1 − lt TV ≤ Cε, if t ≥ u ≥ A (ε) , A (ε) ≥ 1/ε. Therefore, on {lT ∈ Uε (µx ) , t − 1 ≤ τ } , and the region we are considering, we have    t(T − t)  2 (2.28) lt , lt,T  ≤ Cε  T Implementing (2.26), (2.28) and (2.27) into (2.24), we get  2 E eβT lT 2 ; lT ∈ Uε (µx ), ξ > ρ, u < τ (2.29)

 c1 εT +1 (T −t)2 1 Cεβt β T

lt,T 22 dt e E e ; lt,T ∈ Uc2 ε (µx ), Xt ∈ ∂C(ε) (x) . ≤ ρ u We next claim that for y ∈ ∂C(ε) (x)     (T − t)2 2 βT lT 22  Ex e lt,T 2  Ft,T ≥ exp c3 t + β T

(2.30)

on {lt,T ∈ Uc2 ε (µx ), Xt = y} , where Ft,T is the σ-field generated by Xs , t ≤ s ≤ T . Before proving this, we show that (2.29) and (2.30) imply the Proposition 2.42.    2 ZT,β = Ex Ex eβT lT 2  Ft,T   
2 ≥ Ex Ex eβT lT 2  Ft,T ; lt,T ∈ Uc2 ε (µx ), Xt = y y∈∂C(ε) (x)

≥


y∈∂C(ε) (x)

pt (x, y)   βT lT 22  E Ex e  Ft,T ; lt,T ∈ Uc2 ε (µx ), Xt = y . pt (0, y)

Remark that if A (ε) is large enough, we have pt (x, y) /pt (0, y) ≥ 1 for all y ∈ ∂C(ε) (x) and t ≥ A (ε) , if |x| ≥ A (ε) . Therefore,

  
(T − t)2 2 ZT,β ≥ lt,T 2 ; lt,T ∈ Uc2 ε (µx ), Xt = y E exp c3 t + β T y∈∂C(ε) (x) 

  (T − t)2 2 lt,T 2 ; lt,T ∈ Uc2 ε (µx ), Xt ∈ ∂C(ε) (x) . = ec3 t E exp β T

2.4 The collapsed phase for self-attracting random walks

67

Therefore, by (2.29),  2 E eβT lT 2 ; lT ∈ Uε (µx ), ξrε (x) > ρ, u < τ ZT,β c1 εT +1 −c3 t+Cεβt ZT,β −Cu e e dt ≤ C , ≤ ρ ρ u provided ε ≤ ε0 (β) . This proves Proposition 2.42. It remains to prove (2.30). On the prescribed event, the left hand side of (2.30) is     (T − t)2  2 ≥ exp β lt,T 2 Ex e2tβ lt ,µx   Xt = y e−Cεt . T We make a transformation of the path measure switching to the measure (µ ) Px x of a Markov process starting in x having Q-matrix 

1 µx (j)/µx (j) . 2 i,j∈Zd (µ )

Px is absolutely continuous on (Dt , Ft ) with respect to Px x with a density 7 

 t 1 dPx µx (x) 2 ∆ µx (ωs )  exp ds , (2.31) (ω) = (µ ) µx (ωt ) µx (ωs ) 0 dPx x  where ∆ is the discrete Laplacian ∆f (x) = y:|y−x|=1 (f (y) − f (x)). (see e.g. [63], Chapter IV.3). We write now µo (x) = g 2 (x). g satisfies the Euler equation (2.32) 4βg(x)3 + ∆g(x) = λg(x). Multiplying with g(x) and summing over x gives
4 λ = 4β g (y) − 2I(g 2 ) ≥ 2b(β) > 0. x

On the other hand, if we divide 2.32 by g(x), we get t 1  2 ∆ µo (ωs )  ds + 2β lt , µo  = λ. µo (ωs ) 0 The same is of course true if we replace µo by µx . Implementing this into (2.31), implies     (2.33) Ex e2t lt ,µx   Xt = y ≥ Ex e2t lt ,µx  ; Xt = y

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7 =e (µx )

(Xt )t≥0 under Px

λt/2

µo (0) P (µx ) (Xt = y). µo (y − x) x

is ergodic with stationary measure µx . Therefore lim Px(µx ) (Xt = y) = µx (y) .

t→∞

Therefore, there exists to (ε) > 0 such that for t ≥ to (ε) , and all y ∈ ∂C(ε) (x)    Ex e2t lt ,µx   Xt = y ≥ eλt/3 , Therefore, we only have to choose A(ε) ≥ to (ε). This proves (2.30). The Lemmas 2.43, 2.44 and 2.45 now imply Proposition 2.42 in the following way. Given η > 0, we first choose ρ (η) > 0 according to Lemma 2.43 such that (2.34) PˆT (ξr (x) ≤ ρ (η)) ≤ η/3 for all T, x, r. Then we choose ro (η/3) according to Lemma 2.44, so that  √ PˆT τr (0) ≤ r for some r ≥ ro (η/3) ≤ η/3. (2.35)  2 If ε > 0 is given, we choose u (ε, η) ∈ N, u (ε, η) ≥ max A (ε) , ro (η/3) such that C exp [−u (ε, η) /C] ≤ η/3. ρ (η) (C here from Lemma 2.45). If now  def 2 |x| ≥ S (ε, η) = max A (ε) + l (ε) , u (ε, η) then on the complement of the event in (2.35), one has τl(ε) (x) ≥ τu(ε,η)2 (0) , and therefore, according to Lemma 2.45, we have  PˆT lT ∈ Uε (µx ), ξ(ε) (x) > ρ (η) , τ(ε) (x) > u (ε, η) ≤ η/3. (2.36) Combining (2.34),(2.35) and (2.36) we get the statement of Proposition 2.42. In the next section we will need a result which can be proved by a extension of the above argument: Proposition 2.46. limm→∞ lim supT →∞ sup0≤t≤T PˆT (|Xt | ≥ m) = 0. We will not give a proof here which is essentially a repetition of the arguments above. Remark that the supremum over t is outside. The proof of Proposition 2.37 given above shows that lT must have its main weight, up to small probability, close to the starting point. The argument essentially is that if it would be far out, then the path would need some time to reach this place, which would be bad for the Hamiltonian. Completely similar arguments show that uniformly in t we can estimate the probability that the path is at time t far out, just because the path would have to go out from the main bulk out to this place and come back, which again would be bad for the Hamiltonian.

2.4 The collapsed phase for self-attracting random walks

69

2.4.3 Proof of Theorem 2.34 For small enough ε > 0, the Uε (µx ), x ∈ Zd , are all disjoint. We also know from Proposition 2.37 and Proposition 2.46 that for any η > 0 there exist S(η) and m(η) > 0 (not depending on ε !) such that for T ≥ To (ε, η) one has   6 sup PˆT lT ∈ Uε (µx ), |Xt | ≤ m(η) ≥ 1 − η. t≤T

x:|x|≤S(η)

Of course, we can assume S(η) ≤ m(η). We want to prove that for all x ∈ Zd  µo (−x) , (2.37) lim PˆT (lT ∈ Uε (µx )) =   T →∞ µo (y) y for all small enough ε, i.e. ε ≤ εo , εo depending on nothing except d and β. Remark that µo (−x) = µx (0) . From Proposition 2.37 we see that in order to prove (2.37) it suffices to prove that for any x, we have  µx (0) PˆT (lT ∈ Uε (µx )) (2.38) lim =  ˆT (lT ∈ Uε (µo )) T →∞ P µo (0) Given Proposition 2.35, this proves Theorem 2.34, part a). We therefore fix x and take η > 0 such that |x| ≤ S(η) ≤ m(η). We will later on choose some t = t(η), which will not depend on ε (provided always that ε is small enough). We use again the splitting 2

T lT 2 =

t2 (T − t)2 t(T − t) 2 2 lt 2 + lt,T 2 + 2 lt , lt,T  . T T T

If To (ε, η) is chosen large enough (after t = t (η) is chosen), we have the first summand O (η) and replacing the numerator T − t in the third summand by T causes an error of the same size. Therefore, PˆT (lT ∈ Uε (µx )) (2.39) = PˆT (lT ∈ Uε (µx ), |Xt | ≤ m(η)) + O(η)

 2 (T −t) 2 1 = E e2t lt ,lt,T + T lt,T 2 ; lT ∈ Uε (µx ), |Xt | ≤ m(η) + O(η). ZT We want to make some further replacements. Remark % & that if T ≥ To (ε, η) and To (ε, η) is appropriate, we have lT ∈ Uε/2 (µx ) ⊂ {lt,T ∈ Uε (µx )} ⊂ {lT ∈ U2ε (µx )} , and by our tube property, we get !  " lim PˆT (lT ∈ U2ε (µx )) − PˆT lT ∈ Uε/2 (µx ) = 0. T →∞

Therefore, we can replace lT ∈ Uε (µx ) in (2.39) by lt,T ∈ Uε (µx ), making an additional error O (η) . Next, we replace lt,T in lt , lt,T  by µx , which causes an error in the exponent of order tε. Summarizing, we get for εt ≤ 1

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PˆT (lT ∈ Uε (µx )) =

 (T −t)2 2 1 E e2t lt ,µx + T lt,T 2 ; lt,T ∈ Uε (µx ), |Xt | ≤ m(η) ZT

× (1 + O (εt)) + O(η)

 (T −t)2 2 1
E e2t lt ,µx + T lt,T 2 ; lT ∈ Uε (µx ), Xt = y = ZT y:|y|≤m

× (1 + O (εt)) + O(η)   1
 = E e2t lt ,µx   Xt = y ZT y:|y|≤m 

(T −t)2

lT −t 22 T × Ey e ; lT −t ∈ Uε (µx ) (1 + O(εt)) + O(η). The crucial point is that we can choose ε depending on η, and we do it in such a way that εt ≤ η, of course, after having chosen t (η) , which will be done below. In this way, we can replace the error terms above by a summand O (η) , but we have then to take T ≥ To (η, ε) . We explain now, how t (η) has to be chosen. We use the same transformation as in the last section (see 2.33) and get  µx (0) (µx ) λt/2   (Xt = y) e µx (y) P0  E e2t lt ,µx   Xt = y = . Px (Xt = y) If we let t → ∞, we get uniformly in |y| ≤ 2m (η) , Px (Xt = y) = (µ ) Ct−d/2 (1 + o (1)) , and P0 x (Xt = y) = µx (y) (1 + o (1)) . Therefore, if we def

put φ (t) = eλt/2 td/2 /C, we get for t ≥ to (η, m)      E e2t lt ,µx   Xt = y = φ(t) µx (0) µx (y) (1 + O (η)) , uniformly in |y| ≤ 2m (η) . Therefore, after having chosen m (η) , we choose t (η) in this way, and then we have PˆT (lT ∈ Uε (µx ))
φ (t)  µx (0) ≤ ZT



 (T −t)2 2 µx (y)Ey e T lT −t 2 ; lT −t ∈ Uε (µx ) + O(η).

y:|y|≤m(η)

Remark now, that µx (y) = µo (y − x) and  

(T −t)2 (T −t)2 2 2 Ey e T lT −t 2 ; lT −t ∈ Uε (µx ) = Ey−x e T lT −t 2 ; lT −t ∈ Uε (µo ) . As |x| ≤ m (η) , we therefore get for T ≥ To (η, ε (η)) :

2.4 The collapsed phase for self-attracting random walks

71

PˆT (lT ∈ Uε (µx )) 


 (T −t)2 2 φ (t)  µx (0) µo (y)Ey e T lT −t 2 ; lT −t ∈ Uε (µo ) + O(η) ≤ ZT y:|y|≤2m   µx (0)
 1  µo (0) µo (y) ≤ ZT µo (0) y:|y|≤2m

 (T −t)2

lT −t 22 T ; lT −t ∈ Uε (µo ) + O(η) × Ey e  µx (0) ˆ ≤  PT (lT ∈ Uε (µo ), |Xt | ≤ 2m(η)) + O(η) µo (0)  µx (0) ˆ ≤  PT (lT ∈ Uε (µo )) + O(η). µo (0) The above conclusion is for ε = ε (η) , where the latter has been chosen above, but thanks to Proposition 2.35, we can switch back to a fixed ε > 0, small enough, but not depending on η, if T is large enough. In conclusion, we get for a fixed (small) ε  µx (0) ˆ lim inf PˆT (lT ∈ Uε (µo )) . lim sup PT (lT ∈ Uε (µx )) ≤  µo (0) T →∞ T →∞ As the role of x and 0 are interchangeable in the argument, we get the desired relation (2.38). Therefore, part a) of the Theorem 2.34. I will not give the details of part b), as it is by some straightforward modification and extension of the above argument. One has only to introduce another splitting at a time point T − t, to “separated” the endpoint from the main bulk of the empirical distribution. Remark 2.47. From the above proof, it is not difficult to guess what limT →∞ PˆT is, still assuming uniqueness modulo shifts: It is just a mixing of (µ ) the Markovian processes P0 x which are the jump processes with Q-matrix

 1 µx (j) /µx (i) . 2 i,j∈Zd , i =j 5   The mixing is over x and is given by µx (0) µy (0). Therefore, the y limiting measure is not Markovian itself. Remark 2.48. Some last remark about what happens if the Condition 2.33 would fail (a case where I don’t know if it occurs at all). In that case it would be difficult to establish a limiting result and one would have to go into finer asymptotics in large deviation in order to determine the relative weights on the different fibres. This has not been done for the present problem (see

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2 Self-attracting random walks

however [13] for the case of sums of i.i.d. random vectors). However, one can easily get some information: The proof of the tightness essentially applies with only small modifications, and one gets at least tightness for instance of the distribution of the endpoint (and the relative distribution inside each fibre of the Kβ ) without any further assumptions besides b(β) > 0. For the details, I refer to [14].

2.5 A droplet construction for the Wiener sausage A problem which is closely related to the one in the previous section is connected with the classical large deviation result of Donsker and Varadhan for the volume of the Wiener sausage [34]. There is a corresponding result for random walks where the volume of the Wiener sausage is replaced by the number of points visited. I will sketch some of the problems and results in this section without going into technical details. I stick for the moment to the Wiener sausage: So let βt , t > 0, be the standard Brownian motion on Rd , starting in 0. The Wiener sausage is defined by 6 WTa = Ba (βs ), s≤T

where a > 0 and where Ba (x) is the ball with radius a and center x. All results generalize also to the situation where Ba (x) is replace by x + C where C is an arbitrary compact set of positive capacity. The volume of the Wiener sausage is then just its Lebesgue measure VTa = |WTa | . For a random walk (discrete time, say), the natural quantity is the number of points NT visited by the random walk up to time T. It is known that for d ≥ 3, EVTa = κa T + o (T ) , where κa is the Newtonian capacity of Ba (see [67]). For d = 2, one has EVTa ∼ T / log T, and for d = 1, it is of course of order √ T . One is then interested in estimating the probability that VTa is smaller than that. A possibility to measure that is to investigate E (exp [−βVTa ]), β > 0. This quantity appears in a number of problems, for instance in random trapping problems. Consider a Poissonian point process with intensity β > 0 in Rd . Around each point we put a ball of radius a > 0. These balls act as traps. Independently of this point process, we consider a standard Brownian motion. Define the trapping time τ as the first encounter of the Brownian with one of the traps. One is interested in P (τ > T ) , i.e. the (small) probability that no trapping occurs up to time T. P here refers to the joint distribution of the traps and the Brownian motion. To calculate it, we can integrate out

2.5 A droplet construction for the Wiener sausage

73

the Poisson process first. Evidently, there is no trapping if no point of the point process falls into an a-neighborhood of the path of the Brownian, i.e. if the set WTa is trap free. The probability (under the point process) that this happens is exp [−βVTa ] . Therefore P (τ > T ) = E exp [−βVTa ] . Here, on the right hand side, E refers to just taking Brownian expectation. In random media, such a situation is called annealed. This refers to integrating out the trap configuration (the “random environment”) together with the “random walk”, here the Brownian motion. In contrast to this, one can consider the quenched situation. Here one would keep the environment, i.e. the trap configuration fixed, and asks about P (τ > T ) in the T → ∞ limit, where P now refers only to the Brownian. This quantity now depends on the realizations of the traps, so one should write Pω (τ > T ) , ω referring to the trap configurations, and one would then like to have the limiting behavior of this as T → ∞, for almost all trap configurations. I will here entirely focus on the annealed situation. A detailed study of the quenched situation is done in [73]. Regarding E exp [−βVTa ], the classical result of Donsker-Varadhan states: Theorem 2.49. For any β > 0 lim

T →∞

1 log E (exp [−βVTa ]) = ψ(β), T d/(d+2)

where 2/(d+2)

ψ(β) = (ωd β)

d/(d+2)

λd

d/(d+2) 2 d+2 , d 2

ωd being the volume of a ball of radius one and λd is the ground state eigenvalue of the 12 ∆ in the ball with radius one with Dirichlet boundary conditions. There is a similar result for NT in the random walk case (see [35]). In order to understand the result and especially the somewhat strange power of T appearing in this large deviation result, one has first to look at the lower bound. One seeming very crude bound is obtained by confining the Brownian motion inside a ball BrT (0) whose radius rT has to be determined. For such path the volume of the sausage certainly is not larger than the volume of BrT +a (0) which is ωd (rT + a)d . On the other hand, it is well known that   T P (βs ∈ BrT , s ≤ T ) ≥ C exp −λd 2 . rT

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2 Self-attracting random walks

Therefore, we get for any choice of rT :   T E (exp [−βVTa ]) ≥ C exp −βωd (rT + a)d − λd 2 . rT Optimizing over rT one finds that the optimal radius is  1/(d+2) def dT rT ∼ 2λ = ρ(β)T 1/(d+2) , which gives the lower bound in Theodβωd rem 2.49. The difficult part of the theorem is of course the upper bound. It might look somewhat surprising that the above crude argument for the lower bound gives the correct asymptotics, at least in leading order. In order to prove an upper bound one would like to argue roughly as follows:
P (WTa = A) exp [−β|A|] E (exp [−βVTa ]) = A

≤




P (WTa ⊂ A) exp [−β |A|]

A






exp [−β|A| − λ(A)T ] ,

A

where λ(A) is the Dirichlet eigenvalue in A. Of course, the summation does not quite make sense, but it should naturally be understood to run over unions of blocks of side length εa, ε small, of a fixed grid. The main problem is that the sum is running over too many sets. The |A| and λ(A)T are relevant A’s are roughly of diameter T 1/(d+2) where both  typically of order T d/(d+2) . Therefore, there are exp CT d/(d+2) connected A’s which are of the relevant size, so it is clear that one needs some coarsegraining in order to reduce the combinatorial complexity of the summation. It is natural that such a coarse-graining should be possible as the Brownian motion (or the random walk) is smearing out the empirical measure to some extent, so one can believe that one does not really have to sum over so many possibilities. This is also one of the basic ideas of the enlargement of obstacles technique by Sznitman (which works also in the quenched random trap situation not discussed here). I will give no details of these techniques, but will explain in the next section a new approach which has been developed in the “critical” case, recently. Anyway, if one is ready to believe that such a coarse-graining works, one gets ! " log E (exp [−βVTa ]) ∼ exp −T d/(d+2) inf {β|A| + λ(A)} , A

where λ(A) is the Dirichlet ground state eigenvalue of ∆/2 in A, and where log

∼ means that the quotient of the logarithms is going to 1. The variational problem above is a well known one in Mathematical Physics from the beginning of the century, which has been solved independently by Faber and Krahn, who proved that the unique minimizers are the balls. This is closely related to the classical isoperimetric problem, and can be reduced to it.

2.5 A droplet construction for the Wiener sausage

75

A problem in the spirit of the last section is to determine the behavior of the path measure exp [−βVTa ] dP dPˆT,β = ZT for large T. In the formulation as a trapping problem this would be the distribution of the Brownian motion, conditioned not to be trapped up to time T, in the annealed situation. From the Faber-Krahn Theorem it is natural to expect that the paths under this measure are concentrated on balls of radius about ρ(β)T 1/(d+2) . In particular this should mean that the path stays confined within a region of this order. However, even given the techniques to prove the Donsker-Varadhan results, this is far from being evident. The main delicacy is coming from the fact that one has to get some control over certain expectations beyond leading order asymptotics. In this respect the situation is quite similar to the one encountered in the last section, where the different possibilities on the fibers could not be distinguished by leading order asymptotics, too. To see the difficulties in the present problem, consider the event that the Brownian path rushes off through a small tube (of radius 1, say) to a√ distance which is very large compared with T 1/(d+2) , to be specific, say to T , and afterwards settles in √ an optimal ball such far out. This eccentricity gives a contribution of order T to the volume of the sausage, which may look large, but which is negligible when compared with the volume of the optimal √ ball, which is of order T d/(d+2) . The probability for rushing (in time !T , say) √ " through this narrow tube is for the standard Brownian of order exp − T , which may look small, but which is very large compared with the probability that the path does what we expect of it, namely to stay within the optimal  ball, which is exp −CT d/(d+2) . The path could of course do many other things besides just this “tube eccentricity”, and at the outset, it is not clear if one really should believe in this confinement (and had in fact been doubted by experts in the beginning). The problem had first been addressed independently for d = 2 in two papers, the first one by Sznitman [72] and then in [11] for the random walk case. (The first versions of the two papers came out at about the same time.) The confinement has now been proved in a recent paper by Povel [62], which is based on the approach by Sznitman. The results for d ≥ 3 are still not quite as precise as the one for d = 2. Theorem 2.50. There exists a function δ(T ) → 0, as T → ∞, such that a) for d = 2  lim PˆT ∃x ∈ BρT 1/(d+2) (0) : Bρ(1−δ(T ))T 1/(d+2) (x)

T →∞

⊂ WTa ⊂ Bρ(1+δ(T ))T 1/(d+2) (x) = 1

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2 Self-attracting random walks

b) (Povel [62]) for d ≥ 3 :  lim PˆT ∃x ∈ BρT 1/(d+2) (0) : WTa ⊂ Bρ(1+δ(T ))T 1/(d+2) (x) = 1. T →∞

Sznitman’s result contains also the limiting distribution of the centering of the optimal ball, which is not at 0, but which is distributed (after rescaling −1  ) to the normalized ground state eigenfunction of space with ρT 1/(d+2) ∆/2 inside the unit ball. There is no serious doubt that the full result, i.e. also a) and including the limiting distribution, is true in all dimensions, and could probably be proved by some additional efforts. The information on δ(T ) is still very modest. The only information which is known is that one can take some decay of the form T −α , for some α > 0. Bounds for α could be given, but they certainly are not optimal. It seems to be completely out of reach by present day’s methods to get the precise behavior of the boundary, not even for d = 2. An interesting aspect, however, is the proof of such a droplet construction in sup-distance in any dimension.  It is fairly clear that a complete expansion of E e−βVT up to order (1 + o(1)) would be very helpful and desirable for the problem, but this seems to be completely out of reach, too. The methods in [13] do not apply, because VT as a function of the empirical distribution has very bad continuity properties. The best results so far is the one obtained in [11] for the random walk where one takes % & def NT = # x ∈ Zd : Xs = x for some s ≤ T , instead of VT . The rough large deviation result (Theorem 2.49) is exactly the same as in the sausage case. The following sharpening of the statement in Theorem 2.49 is proved (in the random walk case) for all dimensions in [11], provided the variational problem has a rigidity property of the form of Theorem 2.51 below (which I hadn’t known to be proved when writing the paper). The statement is that there exist c1 , c2 , ε (depending on d and β) such that ! " exp −ψ(β)T d/d+2) − c1 (β)T (d−1)/(d+2) ≤ E (exp [−βNT ]) ! " ≤ exp −ψ(β)T d//d+2) + c2 (β)T (d−ε)/(d+2) .

(2.40)

ε can be estimated but presently, there is no hope getting the correct ε. There is a non-rigorous calculation in the physics literature [47], claiming that the correct correction is of the form of the lower bound: E (exp [−βNT ])

! " = exp −ψ(β)T d/d+2) + c1 (β)T (d−1)/(d+2) + o(T (d−1)/(d+2) ) ,

2.5 A droplet construction for the Wiener sausage

77

but this is based on some Gaussian Ansatz for the field of local times, and I do not know how reliable this prediction is. If correct, this would mean that the correction to the volume order large deviations is of surface order. This is a very interesting open problem. One crucial ingredient in all the proofs of results like Theorem 2.50 is an analytic rigidity property of the variational problem, which in our case can be reduced to a rigidity property in the classical isoperimetric problem. This property states that if there is a (nice) subset A in Rd which has as volume that of the ball of radius one, and a surface which is slightly larger, then there exists a ball with radius one which is close in some sense to A. There is a substantial difference between d = 2 and d ≥ 3. For d = 2 such a statement can easily been proved in Hausdorff-distance (with the help of the Bonnesen inequality), but in higher dimension, this evidently cannot be true. In fact, for d ≥ 3, there are sets A with thin spines, these spines having essentially no volume and surface. It is therefore clear that such a rigidity can only be true in some L1 −sense. The following result has been proved by Hall [54]. Theorem 2.51. Let ωd be the volume of the ball with radius 1, and σd its surface. There exist c(d), α(d) > 0 such that for any Borel subset A of Rd with rectifiable boundary ∂A which satisfies |A| = ωd , there exists x ∈ Rd such that |A ∆ B1 (x)| ≤ c(d)(|∂A| − σd )α(d) . It is possible to derive from this a corresponding rigidity result in L1 for the variational problem appearing in the Donsker-Varadhan result 0 / 1 2 2 |∇g| + β 1{g > 0}dx . ψ(β) = inf

g 2 =1 2 This has been done for d = 2 in [11], but the proof there works in all dimensions, given the above theorem. The solutions of this variational problem are unique modulo shifts and given as the ground state eigenfunctions over the ball with optimal radius (β), (i.e. just the usual Bessel function). This is the content of the celebrated Faber-Krahn Theorem. Let F be the set of squares of these optimal profiles. From Theorem 2.51 one can derive (this is not completely evident) Proposition 2.52. There exists δ > 0, and c (β) > 0 such that 1 2 inf{ |  g| + β 1{g2 >0} dx : g 2 dx = 1, inf f − g 2 1 ≥ a} f ∈F 2 ≥ ψ(β) + c(β)aδ . (see Lemma 3.1 in [11]).

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This rigidity property implies in any dimension a corresponding probabilistic property in L1 . Due to bad continuity properties of the Lebesgue measure of the support, even this is not completely evident. However, the main difficulty is to improve this L1 -version to an L∞ -version. I will give a short outline of this in the random walk case. To state the L1 -version, first a compactification is convenient, which is just the usual torus compactification. Fix some multiple of the optimal radius (β), R = 10ρ(β), say. Then we perform the usual periodization on a torus of side-length RT 1/(2+d) , and we scale everything down to a torus of finite side-length R, by replacing the random walk Xt , t ≥ 0, on the torus by ηt = T −1/(d+2) XtT 2/(2+d) , living on LT = T −1/(2+d) {1, . . . , RT 1/(2+d) }d . This is now a process which is running on a torus of fixed size (but with grid which becomes finer and finer). Remark also that the total time for the rescaled process is (R)

def

τ = T d/(d+2) . Next, we consider the ”local times” (R) T (x) (R)

(R)

x ∈ LT . Remark that T

=

T d/(2+d)

1x (ηs )ds, 0

is normalized in the usual sense: (R) T (x)dx = 1,

(R)

if T is extended to the continuous torus [0, R)d by putting it constant on the plaquette of side length T −1/(2+d) . The probabilistic counterpart of the above Proposition 2.52 is the following result which has been proved in [11]. Proposition 2.53. There exists δ > 0 such that (R) lim PˆT,β ( T − F1 ≥ T −δ ) = 0.

T →∞

There is of course no problem to define F on the torus, as the members of F (on Rd ) have as support the balls of radius (β). (R) A consequence of this is that (on our torus) most of the mass of T is concentrated inside a ball of the optimal radius. However, we are not really interested in the torus situation, and it is not clear that the above result should give us anything for the non-compact case. The crucial problem is to boost the result on the torus to a proof that there is no mass outside a ball of radius ρ (β). Once one has proved this confinement property for the random walk on the torus, meaning in particular, that the confining ball most contain

2.5 A droplet construction for the Wiener sausage

79

the starting point, it is evident, that one has also proved that the original problem (on Zd ) has the corresponding confinement property, and therefore, Theorem 2.50 follows, at least the statement in part b). The method how to achieve this had been different in Sznitman’s (and now in Povel’s) papers, and in [11]. Roughly speaking, in [11] it was done by “bare hand”, whereas in [72] and [62] this came out from the enlargement of obstacles technique, together with some other non-trivial considerations. I quickly sketch the main idea used in [11], which might be useful in other contexts, too. It is based on an iterative procedure. The above proposition implies that the total time spent outside a ball of optimal radius ρ is bounded by some τ η = T ηd/(d+2) , η < 1 in the time scale of the rescaled walk. We would like to exclude the possibility that there is any time spent outside. Now assume that the path really spends time T ηd/(d+2) outside the ball, but that it is very pleasant and does this just at the end, and stays confined up to time T ηd/(d+2) − T d/(d+2) . A moment’s reflection shows if the path could only do such a thing, this could easily be excluded. We can separate what the path before and after T ηd/(d+2) − T d/(d+2) is doing (we used such arguments in the lest section). Arguing now anew with the shorter path of length T ηd/(d+2)  T d/(d+2) , we can conclude that under the Gibbs measure, in reality, it would be essentially be confined to a ball of radius  T 1/(d+2) , doing some “really” nasty things just on still a smaller piece, hopefully being pleasant enough to do it at the end such that we can iterate the argument until we have shown that there can be no excursion at all. The trouble is of course that a priori the path has no reason to be so pleasant to do the nasty things just in one piece. There could be many pieces, starting and entering from remote points on the surface of the droplet, and doing all kind of pranks. The argument to get this under control was roughly as follows. One introduces a (finite) number of radii slightly lager than the optimal (β) = r0 < r1 < r2 < . . . rm . Actually the differences ri − ri−1 can be chosen to decay with T . Then one knows from Proposition 2.53 that the total time spent outside r1 by the rescaled walk is at most T dη/(d+2) , η < 1. This can then be boosted to prove that outside r2 there is still less, namely ≤ T dη2 /(d+2) , η2 < η1 . The reasoning roughly is that one can separate what is outside r1 from what is inside and argue as before. One can proceed in this way and prove that outside rm , there is nothing left. The proof in [11] of this inductive cutting of the excursions was quite involving and depended besides Theorem 2.51 on some previous knowledge which was very easy for d = 2, but which was not done in higher dimensions. On the whole, the Sznitman argument was considerably more elegant, but it uses quite special properties. In particular the fact that the ground state eigenvalue is very strongly tied to probabilistic properties was used heavily. The argument sketched above is essentially just a complicated counting argument and might be of use in other problems.

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2.6 Moderate deviation for the Wiener sausage I discuss in this section a recent result obtained together with Michiel van den Berg and Frank den Hollander [6] on what one might call the “critically shrinking Wiener sausage”. It will become clear later on in which sense the situation is “critical”. The interest in this special case is coming from the fact that it is the border-line case where the Donsker-Varadhan behavior, which played a major rˆ ole in the last section, starts to break down, and where the droplet we have discussed in the last section starts to “dissolve” in a way which will become clear. 2.6.1 Introduction and heuristics In the classical Donsker-Varadhan result for the Wiener sausage discussed in the previous section, the main contribution to E (exp [−βVT ]) was coming from paths which stay inside a ball of radius rT = ρ(β)T 1/(d+2) . The “strategy” the path has to follow is somehow trivial: The ball is filled just completely. Even if this is not fully proved in all dimensions, the fact that the trivial lower bound is correct in first order tells us that this is at least up to leading order the correct picture. Consider first a much easier problem namely a Brownian motion, which is conditioned to stay inside a ball of radius T γ . What is the effect on VT of this conditioning? It is well known that under Wiener measure, VT is typically of order κa T, for d ≥ 3, where κa is the Newtonian capacity of a ball with radius a (see [67], [49], there is a logarithmic correction for d = 2). If the Brownian is confined in this ball, the volume can be at most of order T dγ . Therefore, this confinement has trivially a substantial effect on the volume when 0 < γ < 1/d, and it is not difficult to prove that a sausage of the Brownian which is confined to stay inside such a small ball is filling it completely, except near the boundary. Therefore, the volume is (up to smaller order corrections) just the volume of the ball. Let’s look now at the opposite situation where γ > 1/d. In that case, of course, the volume of the ball is much larger than the expectation of the sausage it has when not confined, although for d ≥ 3, and γ < 1/2, confining the Brownian to stay inside BT γ (0) is still a large deviation. It is however not very difficult to see that the confinement in this case has no effect in leading order on the volume of the sausage. i.e. E (VT | WT ⊂ BT γ (0)) = κa T + o (T ) . I don’t know of a reference for this claim, and I don’t want to prove it here, but the reader can easily convince himself of this fact. It is therefore clear that the critical confinement radius which should lead to a sizeable effect on the sausage is of order T 1/d . It is also not difficult to prove that the Brownian motion conditioned to stay inside a ball of radius T 1/d (d ≥ 3) has an expectation of the sausage of order T but smaller than κa T : E (VT | WT ⊂ BT 1/d (0)) = aT + o (T ) with a < κa . Let’s now go back to the (much more difficult problem) to discus E (exp [−βT −α VT ]) . If we proceed with the lower bound explained in the

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last section, the optimal radius to choose will be the one where T −α rTd is of order T /rT2 , i.e. where rT is of order T (1+α)/(2+d) . Now, this radius becomes of order ≥ T 1/d when α ≥ 2/d. It can therefore be expected that for α < 2/d, the Donsker-Varadhan picture stays correct, and   lim T −(d−2α)/(d+2) log E exp −βT −α VT = ψ(β). T →∞

This is indeed true, and has been proved independently in two papers ([10], [70]). The original Donsker-Varadhan approach however does not immediately extend to this situation and has to be refined. For α > 2/d this “Donsker-Varadhan ball picture” breaks down, which should be quite natural given the above discussion. In fact in leading order, the lower bound coming from the Jensen inequality is better than the one coming from the “ball strategy” and turns out to be sharp at least in leading order:   log     E exp −βT −α VT ∼ exp −βT −α E(VT ) ∼ exp −βκa T 1−α . The fact that Jensen is sharp can only mean that it is not “worth” for the Brownian to make any efforts and therefore that the corresponding path measure   dPˆT = exp −βT −α VT dPT /ZT should just be close to ordinary Brownian. This has not been proved, and in view of the discussion given in Section 2.3, one would probably have to prove first that     E exp −βT −α VT ≤ C exp −βκa T 1−α , which has not been done. Anyway, the most interesting case is certainly α = 2/d, where we now have two lower bounds, one coming from Jensen, and the other one from the ball confinement strategy. On the background of the fact that a Brownian which is conditioned to stay in the ball of optimal radius does not fill the ball completely, one would certainly not expect the lower bound to be sharp (in leading order). Somewhat surprisingly, it turns out that the Jensen inequality is sharp for small β, but not for large, where something more interesting is happening, and where also the ball strategy is not the proper thing. This will become clear later. It  turns out  that we better do not start with discussing E exp −βT −2/d VT , but rather with a problem which looks equivalent, but isn’t quite, namely with the probability that VT is small in a range which would correspond to this critical case. It is natural to expect that the discus  sion of E exp −βT −2/d VT is tied to the question of discussing P (VT ≤ bT ),   where b < κa . In fact, we can evaluate E exp −βT −2/d VT in leading order from the evaluation of P (VT ≤ bT ), but not vice versa.

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Theorem 2.54. Assume d ≥ 3. Then for b ∈ (0, κa ) lim T −(d−2)/d log P (VT ≤ bT ) = −I(b)

T →∞

where I(b) = inf

/

1 2 2 ∇g2 : g ∈ H1 (Rd ), g2 = 1, 2



1−e

−κa g(x)2

0 dx ≤ b

> 0. (2.41)

Remark 2.55. a) There is also a version for d = 2. In that case, E(VT ) ∼ κT / log T, where κ is the logarithmic capacity. The Theorem has then to be modified accordingly, i.e.one discusses P (VT ≤ bT / log T ) . b) It is easy to evaluate E exp −βT −2/d VT using Theorem 2.54: ! " lim T −(d−2)/d log E exp −βT −2/d VT = −J(β), (2.42) T →∞

where J is the Legendre transform of I : J(β) = inf {bβ + I(b) : b ∈ (0, κa ]} ,

(2.43)

but not the other way: I is not the Legendre transform of J. This is simply coming from the fact that I is not convex (whereas J is). This will become apparent below. It will also turn out that for small β, the infimum is attained at b = κa , so that for small β one has J(β) = κa β (I(κa ) is of course 0), i.e. the Jensen inequality is sharp in leading order. c) Presently, we are not able to discuss the path measures, for instance discuss the limiting behavior of the distribution of the. end point βT under  P (·|VT ≤ bT ) or under dPˆT = exp −βT −2/d VT dP Z. From the discussion in the last section it should be clear that the measures are living on scale T 1/d , i.e. one would expect that T −1/d βT has under these measures a nontrivial limiting distribution. For PˆT however, there should be a “collapse transition” from small to large β. In the region where J (β) = κa β, i.e. for small β, one would expect diffusive behavior, and only for β large, one would expect a subdiffusivity. However, most probably, there are further complications for d ≥ 5, for reasons which will become apparent in the next section. Nothing on the path measures is proved, and it may be quite difficult. d) The result can easily been extended to more general “sausages” where the ball with radius r is replaced 8 by an arbitrary compact set C with positive capacity, i.e. where WT = s≤T (βs + C). Remark also that the rate function I does depend (via the capacity) on this compact set. This is not done in [6], but it follows by the same method. I first will give an intuitive explanation why the above large deviation principle should hold and why the variational problem looks as it does. Afterwards, I will present in Subsection 2.6.2 the main analytical properties of

2.6 Moderate deviation for the Wiener sausage

83

the variational problem which are quite interesting and surprising. I will not give detailed proofs, but some explanations which hopefully will convince the reader, that the results have to be true. It is clear that the properties of the variational problem should be reflected also in properties of the path measure, but as remarked above, we don’t know how to prove this. Especially, the somewhat strange behavior of the variational problem for d ≥ 5 we will encounter must be reflected in an equally strange behavior of the path measure. I will then give a fairly detailed proof of the interesting probabilistic part of the Theorem 2.54, namely the upper bound in Section 2.6.3. I start with giving a heuristic derivation why the rate function should have the above form. From the discussion previous to the statement of the theorem, it should be apparent that the main contribution to the event {VT ≤ bT } is coming from paths which are staying at distance of order T 1/d from the origin. Furthermore, it should also be clear, that we no longer can expect that the “strategy” of the Brownian being as simple as to just fill a certain region completely, essentially without leaving holes. In contrast, we expect that there remains some porosity, and we have to control the degree of this porosity. The reason that we expect such a porosity is simply coming from the fact that a Brownian motion conditioned to stay inside a ball of radius T 1/d exhibits such a porosity. This can easily be checked (but of course does not prove that such an effect is happening in our problem, too). This porosity is however felt only on a very microscopic scale: It turns out that the holes which are of relevance and are responsible for the porosity have size of order one. What we prove is essentially that the degree of the porosity is tied to empirical distribution at a macroscopic scale (i.e. T 1/d ) deterministically, up to a superexponential estimate. We first rescale the Brownian motion accordingly, by introducing β4t = def −1/d T βtT 2/d , t ≤ τ = T (d−2)/d . As τ is the “correct time scale”, we keep this notation in this way, and use τ always for this. Consider the empirical process τ def 1 Lτ = δ  ds. τ 0 βs By a (weak) LDP, we know that roughly speaking    1 2 P Lτ ∼ f 2 ∼ exp −τ ∇f 2 . 2 It is however not quite clear what Lτ has to do with the volume of the Wiener sausage. Remember that β4t is scaled down by a factor T −1/d = τ −1/(d−2) in space, and therefore      VT = T supp χBaτ −1/(d−2) ∗ Lτ  ,

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where Br (x) is the ball with radius r and center x, Br = Br (0), χA is the indicator function of the set A, and (f ∗ µ) (x) =  f (x − y)µ(dy). There is   + d evidently some trouble as M1 (R )  µ → supp χBaτ −1/(d−2) ∗ µ  depends certainly not continuously on µ, and furthermore depends on τ. We call τ −1/(d−2) the microscopic scale. Let’s look at a small but macroscopic box, i.e. 9d we consider a hypercube Q of side-length ε and center x ∈ Rd : Q = i=1 [xi − ε/2, xi + ε/2). Lτ (Q) measures the relative amount of time, the rescaled Brownian β4t , t ≤ τ , spends inside Q. Evidently, this total amount will usually be cut into many time pieces, the Brownian exiting and reentering the cube. We make a number of very simplifying (false) assumptions: First, we pretend that Q is not a cube, but a torus of the same size with periodic boundary conditions. Next we assume that these many pieces of the Brownian inside Q are just one piece of a Brownian on this torus running up to time Lτ (Q)τ. We will then make this assumption for a collection of Q’s which cover the space and patch things together, but let’s first discuss the problem how much of our Q, which is now a torus, is covered by the (shrinking) sausage. We might hope that the calculation of the expectation is sufficient, and this in fact will turn out to be correct. This may be somewhat surprising as, after all, we are after a large deviation phenomenon, and so we may expect that deviations from expectations will play a rˆ ole. However, we will prove that the deviations of the volume of the microscopic sausage on small macroscopic boxes from its expectation can be estimated on a superexponential scale in τ if the boxes are small (“mesoscopic”). Therefore, we first calculate the expectation of the volume of our (critically shrinking) Wiener sausage (with radius aτ −1/(d−2) ), where the Brownian is running on a torus of side-length ε, and the total time is λεd τ. Lets denote this volume by X. As we have made all kinds of (false) assumptions, we can as well add one more, namely to have the uniform distribution as the starting measure.  EX = dx P ∃s ≤ λεd τ : βs ∈ Baτ −1/(d−2) (x) Q

  / Baτ −1/(d−2) (x), ∀s ≤ λεd τ . = |Q| 1 − P βs ∈ We now chop the time interval [0, λεd τ ) into many pieces of large length K, which we assume not to grow with τ. The probability that the Brownian (with uniform starting distribution) hits Baτ −1/(d−2) (x) in the time slot [0, K) a is Kκ + o(τ −1 ). If the Brownian does not hit the ball in the first interval, εd τ it gets a next change in the second. The conditioning on non-hitting in the first, does not much influence the distribution, as the ball which has to be hit is small anyway. Therefore, we get approximately the same chance for the second slot which is essentially independent of the first one, and so on. We therefore have λεd τ /K

 Kκa d / Baτ −1/(d−2) (x), ∀s ≤ λε τ  1 − d  exp [−κa λ] P βs ∈ ε τ

2.6 Moderate deviation for the Wiener sausage

85

and therefore EX  εd (1 − exp [−λκa ]). We now chop Rd into cubes Qi of the above size, and assume for the moment that Lτ (Qi )  λi εd τ. Then the sausage fills, up to superexponential estimates for the probability not doing so (if the reader believes in what was said above), theQi with a proportion 1−exp [−λi κa ] . Therefore, the total volume covered is i εd (1 − exp [−λi κa ]) . This does all the job on the microscopic scale, and the large deviation we are after is now only a large deviation on the macroscopic scale, i.e. a standard large deviation for Lτ which is governed by the classical DonskerVaradhan LDP. We have to sum over all possibilities for choosing the λi but according to standard wisdom in large deviations, only the maximum counts, and we get / 0 P (VT ≤ bT )  max P (Lτ ∼ f ) : (1 − exp [−κa f (x)])dx ≤ b  exp [−τ I (b)] , /

where I (b) = inf

1 2 ∇g2 : 2



1−e

−κa g 2 (x)

0 dx ≤ b .

That’s it, and there remains only to prove it. I present the real core of the argument in subsection 2.6.3, taking however some (plausible and not too difficult) technical Lemmas for granted. Before starting with it, I want to give some information about the variational problem, which had been quite surprising (at least to us). 2.6.2 Analytical properties of the variational problem I am discussing here the main analytic features of the variational problem (2.41). I will not give detailed proofs, as they partially are quite lengthy, but I will try to explain the main properties. There does not seem to be an explicit solution. It is not too difficult to prove (using standard techniques) that all maximizers of the variational problem are radially symmetric. In principle, one can then discuss the one-dimensional Euler equation, which is just a nonlinear second order differential equation, but this seems not to be of much help. For instance, we have been unable to prove that there is a unique maximizer (modulo shifts), and the problem does not appear to belong to a class which has been treated in the literature. The behavior of I(b) for b ∼ 0 is easy and offers no surprise: The variational problem goes over (after a rescaling) into the variational problem for the classical Donsker-Varadhan situation. It is fairly evident what the

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best way is in which a normed L2 −function can achieve a small value of  2 2 1 − e−κa g(x) dx, best in the sense of having small value of ∇g2 : g just has to be essentially 0 outside some is  small ball. Inside the ball, g then 2 quite large, because of the restriction g 2 (x)dx = 1. Therefore 1 − eκa g (x) is there essentially 1 inside the ball. This means that for small b we have 1 I(b) ∼ inf{   g22 : g2 = 1, | supp(g)| ≤ b}. 2 After rescaling, this leads to Proposition 2.56. For b → 0 I(b) ∼

1 λd (ωd b)−2/d . 2

Much more interesting is the behavior for b ∼ κa . We naturally expect that the relevant functions for the variational problem become flat as b ↑ κ. Following this idea, one expects that we just may expand the exponential: 1 1 − exp(−κa g 2 )  κa g 2 − κ2a g 4 , 2 and replace the restriction by the corresponding restriction on the expanded expression. Implementing the above, we get 2 κ2 (1 − e−κa g )dx  κa − a g 4 (x)dx. 2 This means that for b < κa , b ∼ κa , we should have 0 / 1 κ2a 2 2 4   g2 : g (x)dx = 1, g (x)dx = κa − b . I(b) ≈ inf 2 2

(2.44)

The trouble is that the r.h.s. is 0 for d ≥ 5. This is well known, but I give the proof as it indicates how things should run for d ≥ 5. The claim simply is that for any a > 0, and d ≥ 5 / 0 1 κ2   g22 : g 2 (x)dx = 1, a g 4 (x)dx = a = 0. inf (2.45) 2 2 Here is the sequence, which does the job: We choose a ball with radius 1/n, and over this ball a circular cone of height an . This is gn inside  the ball. We will describe gn outside in a moment. We choose an such that B1/n gn4 (x)dx ≈

a, i.e. an ≈ nd/4 . At the boundary, gn is not quite 0, but this will have no effect on the L4 -norm. The contribution to the L2 -norm from inside the ball is then negligible, and we choose gn outside very flat, producing the necessary L2 -norm. It is clear that we can do that in such a way that this contributes

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87

nothing to the L4 -norm, and also nothing to   g22 (asymptotically). In this way we take care of gn 22 , and   gn 22 is now determined from what is happening inside the ball: |  gn (x)|2 dx ≈ n−d (nan )2 ≈ n−d+2 nd/2 . B1/n

which goes to 0 for d ≥ 5, proving the above claim. This reveals that our approach of expanding the exponential for d ≥ 5 is a failure, and we will come back to this in a moment. For d ≤ 4, this is however the correct procedure, and one can prove the following result: Proposition 2.57. Assume d ≤ 4. Then as b ↑ κa I(b) ∼ 2−

d−2 2

κ−4/d (κa − b)2/d µd , a

where

 a) for d ≤ 3 µd = inf{∇g22 : g ∈ H 1 Rd , g2 = 1, g4 = 1} > 0 b) for d = 4 µd = inf{∇g22 : g ∈ D1 R4 , g4 = 1}.   (For background material about the spaces H 1 Rd and D1 Rd , see [51]) A consequence of this proposition is that for d = 3, 4, I is concave close to κa (of course, the above result does not quite prove this), and has infinite tangent at κa . We come now to the case d ≥ 5. The argument above leading to the conclusion (2.45) does of course not prove that I(b) = 0, simply because the functions in the sequence we have chosen had a high peak inside a small ball, and this peak was important for the result. Remark furthermore, that the whole L2 -norm was “leaking” to ∞ as n → ∞. For the peak inside the ball, the expansion is evidently not the right thing to do, and therefore, (2.45) does not give any immediate indication what the behavior of I (b) should be. In fact I(b) > 0 for all b ∈ (0, κa ). There is however one feature of the above considerations which are important for the behavior of I(b), b ∼ κa , happens for the namely the possibility that L2 is leaking to infinity (which  sequence gn ). To catch this, we apply a trick. For g 2 dx = 1, we have  2 (1 − e−κa g )dx = u if and only if 2 (κa g 2 − 1 + e−κa g )dx = κa − u. The integrand has the advantage that it decays with g 4 if g is small. If therefore L2 -mass of g is wandering to infinity, this is not visible in the integrand, meaning that the integrand would behave continuously, although the L2 -norm would jump. We can therefore try to look at the variational problem forgetting for the moment the g2 = 1 condition, i.e. look at

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2 Self-attracting random walks

1 (ε) = inf{ ∇g22 : 2



2

(κa g 2 − 1 + e−κa g )dx = ε}.

This problem is “well posed”: one can prove that minimizers exist, and the infimum is > 0. In fact, the ε dependence is trivial, and can be obtained by a rescaling (ε) = ε(d−2)/d (1), but it is crucial for this that we have left out the condition g2 = 1. The above equation simply follows from the following observation: If g satisfies  2 (κa g 2 − 1 + e−κa g )dx = 1 then gε (x) = g(ε−1/d x) satisfies  2 κa gε (x)2 − 1 + e−κa gε (x) dx = ε, and ∇gε 22 = ε

d−2 d

∇g22 .

Unfortunately, we have not been able to prove that the variational problem for (1) has a unique minimizer (modulo shifts), and we cannot exclude that there are several minimizers with different L2 -norm, although this does not look very plausible. One can however prove that there are minimizers, which are positive everywhere, and any minimizer has to be rotational symmetric. Let us pretend that there is (modulo shifts) just one or at least that all have the same L2 -norm. If this is not the case, the statement needs some messy but not very important modifications, and the outcome is essentially the same. Let therefore ψ1 be the minimizer for (1) (symmetric around 0, say). If we scale ψ1 to serve for (ε), i.e. take ψε (x) = ψ1 (ε−1/d x), then 2

ψε 22 = ε ψ1 2 . Now, our real problem is to determine  2 1 I(b) = inf{ g2 : g2 = 1, κa g 2 − 1 + e−κa g dx = κa −b}, (2.46) 2 and it looks like that with have not gained very much to calculate (κa − b) because ψκa −b would only be the relevant minimizer if (κa − b)ψ1 22 = 1. However, it turns out that if (κa − b)ψ1 22 < 1, one in fact has I(b) = (κa − b) = (κa − b)(d−2)/d (1). The point is that the variational problem (2.46) does in that case not have a minimizer, because L2 -mass is leaking out to infinity. Therefore, the relevant variational problem is simply the one without the L2 -restriction. This leads to the following conclusion (which is correct regardless of the uniqueness question).

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89

Proposition 2.58. Assume d ≥ 5. Then there exists b0 (d) ∈ (0, κa ) such that for b ∈ [b0 (d), κa ] one has I(b) = (κa − b)(d−2)/d (1). In the case where (κa − b)ψ1 22 > 1, which is true for b small, I(b) has nothing to do with . The L2 -restriction then “deforms” ψ in an essential way. We also know that in this case the variational problem for I(b) has solutions which have L2 -norm 1. For more details and proofs, see [6]. The previous claim that J (β) = κa β (J from (2.43)) for small β now follows easily. From the fact that I (β) has tangent ∞ at β = κa implies that the infimum in (2.43) is attained in β = κa . It is interesting to speculate what the behavior of the variational problem implies for the path measure. It should be evident that for d = 3, 4 and for d ≥ 5 and b small, the paths under P (·|VT ≤ bT ) are living on scale T 1/d , meaning for instance that supE(T −1/d |βT | |VT ≤ bT ) < ∞. T

On the other hand, when d ≥ 5 and b is close to κa , probably the behavior is different. The fact that the variational problem looses mass to infinity can only mean that the path stays “confined on scale T 1/d ” only on part of its life time. For instance, one can imagine that the path first feels the confinement on a fixed proportion of T , and afterwards floats diffusively, but one could also imagine that a more complicated behavior emerges. All this would probably be very difficult to prove. 2.6.3 Proof of the upper bound in Theorem 2.54 I prove here the upper bound, except that I leave some technical lemmas unproved, but I will give some explanations for them. It is convenient to use the usual trivial compactification procedure winding the Brownian motion on a torus. This we do however after having done the rescaling leading to β4s = T −1/d βsT 2/d , s ≤#τ $ = T (d−2)/d . We get VTa = −1/(d−2) T V aτ . We wind the Brownian motion β4s on a torus ΛN of fixed τ

s≤τ

def

size with side length N . By an abuse of notation, we write VτN = Vτaτ but we also often drop the index N. Evidently, we have

−1/(d−2)

,

P (VTa ≤ bT ) ≤ P (VτN ≤ b). To get an upper bound of the left-hand side, we therefore have to bound the right-hand side. The main result to get that is: Proposition 2.59. VτN satisfies a τ -large deviation principle with rate function

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2 Self-attracting random walks

/ 1 2 IN (a) = inf |∇g(x)| dx : g ∈ H1 (ΛN ), 2 ΛN 0    1 − exp −κa g 2 (x) dx = a × ΛN

where H1 (ΛN ) is the usual Sobolev space of once weakly differentiable functions with derivative in L2 (ΛN ). The upper bound in our main Theorem 2.54 follows easily from this proposition. The only thing which remains is Lemma 2.60. limN →∞ IN (a) = I(a) for all a. I will not prove this lemma, which is not difficult. A slight extension of the above proposition also leads to the lower bound in Theorem 2.54. I sketch the argument: Fixing a (large) number R, we have (on Rd )

 P (VT ≤ bT ) ≥ P

VT ≤ bT, sup |βt | ≤ RT 1/d . t≤T

For the rescaled Brownian, the second event on the right-hand side is  4  ole whether supt≤τ βt  ≤ R. If we choose N > R, then it doesn’t play any rˆ the probability above is calculated for the torus Brownian motion or for the N unconfined one. A slight #extension  of  the $proposition above gives for Vτ , 4  conditioned on the event sup βt  ≤ R a τ -LDP with rate function t≤τ

def

/ 1 2 |∇g(x)| dx : supp (g) 2 ΛN 3  d ! " R R −κa g 2 (x) ⊂ − , , 1−e dx = a . 2 2

IR (a) = inf

Then lim inf T →∞

1

log P (VT ≤ bT )

1 log P VT ≤ bT −(d−2)/d

T −(d−2)/d

≥ lim inf T →∞

+ lim inf T →∞

T

1 T −(d−2)/d

= −IR (b) + lim inf T →∞

    sup |βt | ≤ RT 1/d  t≤T

 1/d log P sup |βt | ≤ RT t≤T

1 T −(d−2)/d

log P

 sup |βt | ≤ RT

1/d

.

t≤T

The lower bound in Theorem 2.54 then follows by letting R → ∞. I don’t give the details which are not very interesting.

2.6 Moderate deviation for the Wiener sausage

91

I will give the proof of the Proposition 2.59 in some details. From its form, it is clear that we should get it by a kind of contraction principle. It seems however impossible to get that directly, and we use an approximation procedure. For the rest of this chapter, the torus ΛN is fixed. We usually drop N in the notation. We also drop the tilde in β4s , and just write βs for this rescaled Brownian motion. Time is always running up to τ. Here is an outline of the procedure: A) We first approximate Vτ (= VτN ) by its conditional expectation Eε (Vτ ) = E(Vτ | {βiε }0≤i≤τ /ε ), where ε is a parameter > 0. We prove that the difference between Vτ and Eε (Vτ ) is negligible in the ε → 0 limit. This is done by an application of a concentration inequality of Talagrand. B) We represent Eε (Vτ ) as a functional of the empirical distribution Lε,τ =

τ /ε ε
δ(β ,β ) . τ i=1 ε(i−1) εi

According to one of the very basic large deviation results of Donsker and Varadhan, Lε,τ satisfies for fixed ε a strong LDP (on the torus). We still will need some further approximations to get the dependence of Eε (Vτ ) on Lε,τ in a suitable form, but essentially based just on this basic LDP for Lε,τ , we get via a contraction principle a LDP for Eε (Vτ ). C) We finally have to perform the ε → 0 limit. We now already know that Vτ is approximated by Eε (Vτ ). It therefore will suffice to have an appropriate transition for the variational formula. We write Xτ,ε = {βiε }1≤i≤τ /ε . (For notational convenience, we always assume that τ /ε is an integer). We denote by Pε and Eε the conditional probability and expectation with respect to Xτ,ε . The first main step (A) is to prove that Vτ is well approximated by Eε (Vτ ) in the following sense: Proposition 2.61. For all δ > 0 we have lim lim sup

ε→0 τ →∞

1 log P (|Vτ − Eε (Vτ )| ≥ δ) = −∞. τ

Proof. The proof is based on Talagrand’s concentration inequalities. We denote by mτ,ε the median of the distribution of Vτ under the conditional law Pε . Furthermore, let Wi , 1 ≤ i ≤ τ /ε, be defined by 6 Baτ −1/(d−2) (βs ). (2.47) Wi = s∈[(i−1)ε,iε]

Evidently, the Wi are, conditionally on Xτ,ε , are independent random closed subsets of ΛN , and we have

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2 Self-attracting random walks

   6τ /ε  Vτ =  Wi  . i=1 Let S be the set of closed subsets of ΛN . The mapping d : S × S → [0, ∞), d(A, B) = |A∆B|, defines a pseudometric on S. We equip S with the Borel field S generated by this pseudometric. Pε then defines a product measure on (S, S)τ /ε , which, by an abuse of notation, we denote by Pε , too. We apply one of Talagrand’s concentration inequality to the function V : S τ /ε → [0, ∞), defined by  6  τ /ε   V (C) =  Ci  , C = {Ci }. i=1

Evidently, V is Lipshitz in the sense that

|V (C) − V (C )| ≤

τ /ε


|Ci ∆Ci | .

i=1

$ # A = C ∈ S τ /ε : V (C) ≤ mτ /ε .

Let

The distribution of V under Pε has no atoms. Therefore, we have Pε (A) = 12 . From Theorem 2.4.1 of [74], we have Eε (exp [λf (A, {Wi })]) ≤ 2

τ /ε :

Eε (cosh(λ |Wi ∆Wi |)),

i=1



where f (A, {Ci }) = inf {Di }∈A i d(Ci , Di ) and {Wi } is an independent copy of {Wi }. From the Markov inequality, we therefore get Pε (f (A, {Wi }) ≥ δ) ≤ 2 inf e−λδ λ>0

τ /ε :

Eε (cosh(λ |Wi ∆Wi |))

i=1

= Φτ,ε (δ), say. & % Arguing similarly with A = C ∈ S τ /ε : V (C) ≥ mτ /ε , we get Pε (|Vτ − mτ,ε | ≥ δ) ≤ 2Φτ,ε (δ). Remark now that |Vτ | is bounded by |ΛN |. Therefore

 δ δ |Eε (Vτ ) − mτ,ε | ≤ + 2 |ΛN | Pε |Vτ − mτ,ε | ≥ . 3 3 Using this, we have

(2.48)

2.6 Moderate deviation for the Wiener sausage

93

    δ δ δ Pε (|Vτ − Eε (Vτ )| ≥ δ) ≤ 2Φτ,ε + I Pε |Vτ − mτ,ε | ≥ ≥ 3 3 6 |ΛN |  



 δ δ δ ≤ 2Φτ,ε + I 2Φτ,ε ≥ , 3 3 6 |ΛN | where I[.], denotes the indicator function of an event. Using this inequality, we get

  δ 6 |ΛN | P (|Vτ − Eε (Vτ )| ≥ δ) ≤ 2 1 + E Φτ,ε . δ 3 In order to prove the Proposition, it therefore suffices to show that lim lim sup

ε→0 τ →∞

1 log E (Φτ,ε (δ)) = −∞ τ

(2.49)

holds for all δ > 0. We actually prove more, namely lim lim sup

ε→0 τ →∞

1 log Φτ,ε (δ)∞ = −∞. τ

(2.50)

To estimate Φτ,ε (δ) we will take λ = αε−1 τ with 0 < α ≤ 1 in Eε (cosh(λ |Wi ∆Wi |)). Remark that cosh(αb) ≤ 1 + α2 exp(b), if 0 < α ≤ 1 and b > 0. If x ∈ ΛN , we write Ex,ε for the expectation under a Brownian bridge on the ΛN -torus, i.e. a Brownian motion (βs )0≤s≤ε starting at 0 and conditioned to be at x at time ε. It is evident that the volume for the sausage of such a Brownian bridge on the torus is stochastically smaller than the corresponding sausage of a Brownian bridge on Rd . We then have Eε (cosh(α(τ /ε) |Wi ∆Wi |))  " !  −1/(d−2)   ≤ 1 + α2 Eβεi −βε(i−1) ,ε (exp (τ /ε) W aτ (ε)

2

,

8 where W a (t) = s≤t Ba (βs ). As remarked above, we can replace the right hand side in the above inequality by the corresponding quantity for the standard Brownian motion, which has the advantage that we now can use the standard rescaling properties. Using these, we get "   ! −1/(d−2)   (ε) Ex,ε exp (τ /ε) W aτ  "  !   ≤ Eτ∞1/(d−2) x,ετ 2/(d−2) exp ε−1 τ −2/(d−2) W a (ετ 2/(d−2) ) , where P ∞ , E ∞ refer to the Brownian on Rd . According to the Lemma 2.62 below, we see that there is a τo (ε, N ) such that for all τ ≥ τo (ε, N ), all N , and all x ∈ ΛN we have  "  ! −1/(d−2)   ∞ exp (τ /ε) W aτ Ex,ε (ε) ≤ C.

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2 Self-attracting random walks

We therefore get τ /ε :

Eε (cosh(λ |Wi ∆Wi |)) ≤

i=1

τ /ε :

 (1 + α2 C 2 ) ≤ exp (τ /ε)α2 C 2 .

i=1

Implementing it into (2.48), we get ! ε" ε Φτ,ε (δ) ≤ 2 exp −δα + C 2 α2 τ τ and choosing now α small enough (2.50) follows, and therefore the Proposition 2.61 is proved. Lemma 2.62. There exists a constant C with 

 1 ∞ ≤ C. sup Ex,t exp |W a (t)| t t≥1,|x|≤t I will not give a proof of this. For the unconditioned Brownian motion, this follows from estimates in [7]. The lemma states that the situation does not change much if we condition the Brownian to end in a point which is away from the starting point at maximum t. Although this is a large deviation for the Brownian, it is evident that this increases the sausage at maximum to something of order t, and so the statement of the Lemma looks plausible. It is not difficult to prove if by chopping time into small pieces. We have finished the first part (A) of the proof, and we come to (B). During the proof of this part, we keep the parameter ε completely fixed. We denote by ps the transition densities for the Brownian motion (on the torus ΛN , but as usual, we drop the N in the notation). For y, z ∈ ΛN we define qbε (y, z) = P (∃s ≤ ε with βs ∈ Bb (0)|β0 = y, βε = z), ε and by an abuse of notation qτε (y, z) = qaτ −1/(d−2) (y, z) where a is the radius of the original sausage. We also set for y, z = 0 ε ps (y)pε−s (z) ds . ϕε (y, z) = 0 pε (z − y)

It is evident (see below) that Eε (Vτ ) can be expressed with the help qτε (y, z) and the empirical measure Lε,τ , and we therefore easily get a LDP, except for the problem that qτε (y, z) still depends on τ. We don’t like this τ -dependence. The basis for being able to remove it is the following technical result. Lemma 2.63. a) Let b < b1 < N/4. Then

sup x,y ∈B / b1

qbε (x, y)

≤C

b b1

d−2

2.6 Moderate deviation for the Wiener sausage

95

b) For any ε, b > 0 we have lim

sup

τ →∞ y,z ∈B / b (0)

|τ qτε (y, z) − κa ϕε (y, z)| = 0,

where κa is the Newtonian capacity of the ball with radius a. a) is rather evident and easy to prove. Remember that ε is fixed. The claim is that if the starting and the end point of the bridge are sufficiently for away from the ball to be hit, then there is only a small chance for this hitting. The exact form of the estimates comes easily from standard estimates of hitting probabilities. b) is more delicate. From scaling, one sees that qτε (y, z) is in fact of order τ. The bridge has a chance to hit the small ball only if it already gets close to it. ϕε (y, z) measure the expectation of the total time, the bridge spends in the vicinity of the ball. This quantity has to be multiplied with the capacity of the ball, which is κa /τ. For details, see [6]. We now perform the approximation of Eε (Vτ ). We first approximate Vτ by cutting out small holes around the points βiε : Fix b > 0 and define  Wib = Wi \ Bb (β(i−1)ε ) ∪ Bb (βiε ) ,  6  τ /ε Kτ −1/(d−2)  . VτK =  Wi  i=1

and set

Evidently, we have cut out at maximum τ /ε times the volume of a ball of radius Kτ −1/(d−2) . Therefore   Vτ − VτK  ≤ cε−1 K d τ −2/(d−2) , (2.51) and therefore the difference is negligible for our purpose. The cutting is convenient, because we can invoke then the Lemma 2.63 which helps to expand log(1 − q) linearly in q just by −q.

 6τ /ε −1/(d−2) Eε (VτK ) = dx 1 − Pε (x ∈ / WiKτ ) i=1

ΛN

= ΛN



=



dx 1 −

τ /ε ! :

1 − Pε (x ∈ WiKτ

−1/(d−2)

"



) 

(2.52)

i=1

   −1/(d−2) τ log 1 − qτε,Kτ dx 1 − exp (z − x, y − x) Lε,τ (dy, dz) , ε

ΛN

/ Bb (0) and 0 otherwise. We are therefore where qτε,b (z, y) = qτε (z, y) if z, y ∈ naturally led to the investigation of mappings M+ 1 (ΛN × ΛN ) → [0, ∞)

96

2 Self-attracting random walks def

  ε,b dx 1 − exp −βτ qτ (z − x, y − x)µ(dy, dz) .



Φτ,β,b (µ) =

ΛN

Then, we get the sandwiching Φτ,(1+δK )/ε,Kτ −1/(d−2) (Lε,τ ) ≤ Eε (VK ) ≤ Φτ,1/ε,Kτ −1/(d−2) (Lε,τ ), with δK → 0 for K → ∞. This follows from Lemma 2.63, part a). With the same lemma, we also see that we can replace Kτ −1/(d−2) with a fixed (small) value b : 1 1 1Eε (VK ) − Φτ,1/ε,b (Lε,τ )1 ≤ δ1 (τ, K, b), (2.53) ∞ where limb→0 lim supK→∞ lim supτ →∞ δ1 (τ, K, b) = 0. (Of course, we just estimate |exp [−ξ] − exp [−η]| ≤ |ξ − η|). Instead of spelling out the details for the above estimate which are easy, I want to give a comment on what is going on: The one reason that we did cut out “only” Kτ −1/(d−2) -holes was that we wanted to use a very crude bound of the total amount cut out. However, having now arrived at an approximation by Φτ,1/ε,Kτ −1/(d−2) (Lε,τ ), we want to cut bigger holes (for the procedure done in a moment). The reader might wonder that this is possible. The essential point is that the chance to hit the aτ −1/(d−2) -ball in an interval of length ε is very small anyway. Of course, we have to have to know this probability because we multiply it by τ in Φ, but it is not very important if our starting point is only close to the aτ −1/(d−2) ball (still macroscopic) or “very close” (i.e. on scale τ −1/(d−2) ). This region between “close” and “very close” is negligible, due to our lemma, essentially because we have the x-integration in the end. Define now

  def b Φ∞,β,b (µ) = dx 1 − exp −βκa ϕε (y − x, z − x)µ(dy, dz) , ΛN

where ϕbε (x, y) is ϕε (x, y) if x, y are both outside Bb (0), and 0 otherwise. Lemma 2.63 b) now gives 1 1 1Φ∞,1/ε,b (Lε,τ ) − Φτ,1/ε,b (Lε,τ )1 ≤ δ2 (τ, b), (2.54) ∞ where limτ →∞ δ2 (τ, b) = 0 for all b. Combining now (2.53) and (2.54), we get, by letting τ → ∞, K → ∞, and finally b → 0 (in this order): 1 1 lim 1Φ∞,1/ε,0 (Lε,τ ) − Eε (Vτ )1∞ = 0. τ →∞

Φ∞,1/ε,0 (µ) is continuous in µ, and therefore, we get the following large deviation principle for Eε (VT ) (ε arbitrary > 0, but fixed), which is based on a (strong) LDP for bivariate chains, stated after the proposition.

2.6 Moderate deviation for the Wiener sausage

97

Proposition 2.64. {Eε (Vτ )}τ >0 satisfies a τ -LDP with rate function # $ def Jε (b) = inf Iε(2) (µ) : µ ∈ M+ 1 (ΛN × ΛN ), Φ∞,1/ε,0 (µ) = b . (2)

Here Iε (µ) is the rate function of the LDP for Lε,τ on M+ 1 (ΛN ×ΛN ) which is just 

dµ (2) Iε (µ) = log dµ, d(µ1 ⊗ πε ) if µ1 = µ2 , µi being the margins of µ, and ∞ otherwise. πε is the transition kernel of the Brownian on the torus on a time interval ε. The proposition follows from the considerations explained above, a contraction principle (see [43], Ch. III.5) and the following result Theorem 2.65 (LDP for bivariate Markov chains). Let ξi , i ∈ N be a Markov chain, taking values in some Polish space S with transition densities p (x, y) with respect to a stationary measure π which satisfy 1/C ≤ p (x, y) ≤ C. Consider the bivariate empirical distribution 1
δ(ξ ,ξ ) . n i=1 i−1 i n

def

L(2) n =  (2) Then Ln

satisfies a (strong) N -LDP in M+ 1 (S × S) with rate function / def

I (2) (µ) =

log

dµ d(µ1 ⊗p) dµ

∞

if µ1 = µ2 , if µ1 = µ2

where µ1 , µ2 are the two marginals of µ. µ1 ⊗ p is the measure µ1 (dx) p (x, y) π (dy) . For a proof of this, see [43], Theorem IV.3. We come now to the last step C) of the proof of Proposition 2.59. Up to now, we have a LDP for Eε (Vτ ), and we know that this quantity approximates the one we are interested in. We therefore only have to prove that the rate function approximates the right one. There is one delicacy. The rate function we have for fixed ε is a rate function of the bivariate chain. It is well known, that the rate function of the univariate discrete time ε-gap chain approximates the one for the Brownian motion as ε → 0, and the rate function of the univariate discrete chain is the projection of the bivariate one. In our case, the function really depends on the bivariate chain. It however

98

2 Self-attracting random walks

turns out that for small ε, the bivariate chain is essentially determined by the univariate one, up to a superexponential decay. For µ ∈ M+ 1 (ΛN ), we write I(µ) for the standard large deviation rate function for the empirical distribution of the Brownian motion: I(µ) =  2 1 2 |∇g| dx, g (x) = µ(dx)/dx if µ is absolutely continuous, and its density 2 + We also denote by is in H1 and I(µ) = ∞ otherwise. # $ Iε : M1 (ΛN ) → [0, ∞] (2)

(2)

the projection of Iε : Iε (ν) = inf Iε (µ) : µ1 = ν . We collect some basic facts about these entropies which have been proved by Donsker and Varadhan or are simple consequences of their results: Lemma 2.66. Let (πt )t≥0 be the Brownian semigroup. Then for all ν, µ ∈ M+ 1 (ΛN ) we have  a) Iε (ν) = − inf u∈D+ log πuε u dν, where D+ is the set of positive measurable functions which are bounded and bounded away from 0. b) t → It (ν)/t is non-increasing with I(ν) = limt→0 It (ν) t .  c) ν − νπs TV ≤ 8 Is (ν) for s > 0 t > 0. d Is (νπt ) ≤ Is (ν) for s, (2)

e) µ − µ1 ⊗ πs TV ≤ 8 Is (µ) Proof. a) This is Theorem 2.1 of [33], combined with Lemma 2.1 of [32]. b) Let u ∈ D+ and s, t > 0. Then πs (πt )u πt u πs+t u dν = log dν + log dν ≥ −Is (ν) − It (ν). log u πt u u Therefore Is+t (ν) ≤ Is (ν) + It (ν). Hence, It (ν)/t is non-decreasing. The fact that limt→0 It (ν)/t = I(ν) is Lemma 3.1 from [32]. c) This √ is Lemma 4.1 of [32]. (The function φ used there is easily seen to be ≤ 8 x). d) follows from the convexity of Is . e) Let P µ (x, dy) be a transition kernel on ΛN with µ = µ1 ⊗ P µ . Then µ − µ1 ⊗ πs TV ≤ µ1 (dx) P µ (x, ·) − πs (x, ·)TV . By Theorem 4.1 of [28], we have 

P µ (x, ·) − πs (x, ·)v ≤ 8

k(P µ (x, ·)|πs (x, ·)),

where k is the usual Kullback–Leibler information, i.e. k(γ|σ) = log(dγ/dσ)dγ. Therefore

2.6 Moderate deviation for the Wiener sausage

µ − µ1 ⊗ πs TV ≤ 8

99

 µ1 (dx) k(P µ (x, ·)|πs (x, ·))

7

≤8

µ1

(dx)k(P µ (x, ·)|π

 (2) s (x, ·)) = 8 Is (µ).

Next, we need an approximation of our functions Φ∞,1/ε,0 , for which we had derived a LDP by the Proposition 2.64, by the simpler functions Ψε : M+ 1 (ΛN ) → [0, ∞), defined by 

 κa ε Ψε (ν) = dx 1 − exp − ps (y − x)ν(dy) . ε 0 Lemma 2.67. For any K > 0 lim

ε→0

sup (2)

µ: 1ε Iε (µ)≤K

  Φ∞,1/ε,0 (µ) − Ψε (µ1 ) = 0.

Proof. We have Ψε (µ1 ) = Φ∞,1/ε,0 (µ1 ⊗ πε ), and therefore   Φ∞,1/ε,0 (µ) − Ψε (µ1 )   = Φ∞,1/ε,0 (µ) − Φ∞,1/ε,0 (µ1 ⊗ πε )    κa  dx ≤ ϕε (y − x, z − x) (µ(dy, dz) − µ1 ⊗ πε (dy, dz)) ε  Λ ×Λ N N κa dx ≤ ϕε (y − x, z − x) |µ − µ1 ⊗ πε | (dy, dz) ε ΛN ×ΛN = κa µ − µ1 ⊗ πε T V . The Lemma follows now from Lemma 2.66 e). Next, we define Γ : L+ 1 (ΛN ) → [0, ∞) by Γ (f ) = dx [1 − exp (−κa f (x))] . Lemma 2.68. For any K > 0      dν  Γ sup lim − Ψε (ν) = 0.  ε→0 ν: 1 I (ν)≤K dx ε ε

(Remark that if Iε (ν) is finite, then dν  dx) Proof.      Γ dν − Ψε (ν)   dx

100

2 Self-attracting random walks

 



  κa ε κa ε dν  ≤ dx exp − ds ps (y − x)ν(dy) − exp − ds (x)  ε 0 ε 0 dx    κa ε κa ε  νπs dν ≤ dx (x) − (x) = ds  ds νπs − νTV . ε 0 dx dx  ε 0

Now, for s ≤ ε νπs − νTV ≤ νπs πε − νπs TV + νπs+ε − νTV   ≤ 8 Iε (νπs ) + 8 Iε+s (ν). Now Iε (νπs ) ≤ Iε (ν) by Lemma 2.66d). s) ≤ 2Iε (ν) by Lemma 2.66b). Furthermore, Iε+s (ν) ≤ 2εIε+s (ν)/(ε√+ √ Therefore, we get νπs − νTV ≤ 8(1 + 2) Kε if Iε (ν) ≤ Kε. Using this, the Lemma follows. We can now finally finish the proof of Proposition 2.59. Consider a continuous bounded function f : R → R. Then  1 lim log E eτ f (Vτ ) τ →∞ τ 1 = lim lim log E (exp [τ f (Eε (Vτ ))]) (Proposition 2.61) ε→0 τ →∞ τ 0 / 1 (2) = lim sup f (Φ∞,1/ε,0 (µ)) − Iε (µ) (Proposition 2.64) ε→0 µ ε 0 / 1 (2) sup = lim lim f (Φ∞,1/ε,0 (µ)) − Iε (µ) K→∞ ε→0 (2) ε Iε (µ)≤εK 0 / 1 (2) = lim lim sup f (Ψε (µ1 )) − Iε (µ) (Lemma 2.67). K→∞ ε→0 (2) ε I (µ)≤εK ε

(2)

We now use that Iε is the projection of Iε , namely # $ Iε (ν) = inf Iε(2) (µ) : µ1 = ν . Therefore

0 /  1 1 τ f (Vτ ) f (Ψε (ν)) − Iε (ν) = lim lim sup lim log E e τ →∞ τ K→∞ ε→0 Iε (ν)≤εK ε 0 /  dν 1 = lim lim sup f Γ − Iε (ν) K→∞ ε→0 Iε (ν)≤εK dλ ε /  0 dν = sup f Γ − I(ν) , dλ ν

the second equation by Lemma 2.68. This proves now the Proposition 2.59 by applying the “inverse” of Varadhan’s lemma, also called Bryc’s lemma (see [43], p. 33).

2.7 Remarks on the polaron problem

101

2.7 Remarks on the polaron problem A problem similar in spirit to the considerations in this chapter is connected with the so-called polaron problem. The questions on path measures are mathematically open, but I explain what is known and what is conjectured in this interesting problem. The physical problem is coming from a quantum mechanical discussion of a charged particle, e.g. an electron, which is moving in a crystal whose lattice sites can be polarized. The electron is then dragging around it a cloud of polarized lattice points which will influence its behavior. In particular, the electron moves as having a different mass. This is the so-called effective mass. I will not discuss the physical background for which I refer to the Lectures of Feynman [39]. Feynman gave a path integral formulation of the problem, and questions about the effective mass can be formulated in terms of a path measure obtained from the Brownian path by a self-attracting interaction. We are describing actually only the Fr¨ ohlich polaron (after the solid-state physicist H. Fr¨ ohlich). There are other ones which not all fit into this framework. Here is the path measure, which has two parameters β, λ > 0 :

 T T −λ|t−s| 1 βλ e def P (dω) . P-T,β,λ (dω) = exp ds dt ZT,β,λ 2 0 |ωt − ωs | 0 P is the law of the three-dimensional Brownian motion. The parameter β is not of importance and can be scaled away. So we put β = 1. The above form is actually not exactly the one given in Feynman, but it follows by some trivial rescaling. The parameter λ is 1/α2 , where α is the “physical” coupling parameter between the electron and the lattice points. We are interested in the case λ → 0, which corresponds to the strong coupling limit for the physical problem. First remark that the interaction is self-attracting: The new measure favors paths which, at least on short time scales, are clumping together. (It should also be remarked that the integral is well defined in three dimensions, despite of the Coulomb singularity). An important feature is that the interaction decays exponentially in |t − s| , so that for fixed λ, the interaction is essentially short range in time. The effect is that the path measure behaves essentially diffusive (for fixed λ), i.e. after Brownian rescaling, P-T,λ converges to the Brownian motion with a rescaled diffusion coefficient D (λ) > 0, i.e. (D(λ)I) , lim P-T,λ ρ−1 T =P

T →∞

def

(Σ) where P√ is the Brownian motion with covariance matrix Σ, and ρT (ω) = ω (T ·) / T , as usual. There does not seem to exist a proof of this for the above model. Spohn ([68]) proved it under some smoothness assumption on the interaction (not including the Coulomb singularity, however), but I think there can be no serious doubt that the statement is correct, despite the fact

102

2 Self-attracting random walks

that there had been speculations in the physics literature about a roughening transition. An interesting problem is to determine the path properties of the above path measure for λ ∼ 0, and in particular to determine D (λ) , but this is mathematically open. The diffusion constant D (λ) is directly related to the effective mass (see [68]). An easier problem is to determine the (rough) behavior of the partition function ZT,λ , and this has been done in a celebrated result by Donsker and Varadhan. The argument roughly is that for λ small, the interaction, despite being short range, gets more and more smeared out, and one might guess that for λ → 0 (after T → ∞), the behavior of Z does not much deviate from the situation where one would take the mean-field model with a Hamiltonian T 1 T 1 . ds dt T 0 |ωt − ωs | 0 This was the content of the Pekar-conjecture and is true: Theorem 2.69 (Donsker-Varadhan (see [36]).  

 T 1 T 1 1 1 lim lim log ZT,λ = lim log E exp ds dt T →∞ T λ→0 T →∞ T T 0 |ωt − ωs | 0 / 0 dx dy 2 1 2 = sup g (x) g 2 (y) − ∇g2 . 2

g 2 =1 R3 R3 |x − y| The second equation is in the spirit of the large deviation arguments we have used above: The mean-field Hamiltonian can be written as T 1 1 T 1 =T LT (dx) LT (dy) , ds dt T 0 |ωt − ωs | R3 R3 |x − y| 0 from which the last equality “follows” by the Donsker-Varadhan Theorem 2.17 stated in the introduction 2.1, but there are a number of technical difficulties, e.g. the singularity. However, the the main difficulty was to prove the first equation for which Donsker and Varadhan developed their “level-3” large deviation principles. I will not go into that here. The variational problem above had been discussed in a celebrated paper by Lieb [50] who proved that there are unique maximizers modulo shifts. From this result it remains somewhat unclear what to expect for the path measure. It should however be clear that the problem is in some way related to the mean-field path measure:

 T 1 1 T 1 def mf P (dω) . PT (dω) = mf exp ds dt T 0 |ωt − ωs | ZT 0 The following “result” is then in complete analogy to our Theorem 2.34:

2.7 Remarks on the polaron problem

103

Conjecture 2.70. lim P-Tmf L−1 T =

T →∞



g (x) δθx g2 dx  , g (x) dx

where g is the unique L2 -normed, positive, rotational symmetric solution of the variational problem in Theorem 2.69. (By an abuse of notation, θx g 2 stands for the measure on R3 with this density). This has also not been proved, but in my view there can be no serious doubt that it is correct. A proof could probably be given along the lines of the proof of Theorem 2.34, with some additional technical problems coming from the Coulomb singularity. Given this “result”, one should believe that our real path measure P-T,λ looks at least on scales of order 1/λ such that the local empirical measures are close to some θx g 2 . Based on this, Herbert Spohn in [68] gave a heuristic derivation of what D (λ) should be. This heuristic argument is based on a number of very simplifying assumptions. First chop the time axis into pieces def Ik = [(k − 1) η, kη) of length 1  η  1/λ, 1 ≤ k ≤ T /η, and consider the empirical distribution on each of these time slots: def 1 Lk,T = δω ds. η Ik s The basic assumption is that these local empirical measures are close to some shift of g 2 : Lk,T ∼ g 2 (· − θk ) , which is certainly plausible if one believes in the above Conjecture 2.70. Spohns argument is now based on the following further assumptions: • The only relevant information are these θk and the fluctuations of Lk,T around g 2 (· − θk ) are not playing any rˆ ole. • The fact that the end points of the Brownian motion at the end of one of the time slots is the same as at the beginning of the next slot is not having any influence. • The a priori distribution of the θk is “uniform distribution” on R3 . • The diffusion constant can be evaluated (in the λ → 0 limit) by forgetting anything except the θk , and expand the Hamiltonian in terms of these parameters. This evidently leads to a Gaussian theory for the sequence (θk )k≥1 . The rest of the argument is then plain sailing, but it goes without saying that a justification of the above assumptions is very far from obvious, and the reader will probably have serious doubts that the answer obtained in this way is correct (as had I, first). First, we can write the Hamiltonian as

104

2 Self-attracting random walks

T /η λ
1 ds dt e−λ|t−s| 2 |ωs − ωt | Ik Il k,l=1

T /η 1 λ
−λη|l−k| e ds dt = 2 |ω − ωt | s Ik Il k,l=1

≈

T /η 1 λη 2
−λη|l−k| g 2 (x − θk )g 2 (y − θl ) , e dx dy 2 |x − y| R3 R3 k,l=1

where we have used the basic assumption that we can switch from the local empirical measures to the g 2 (x − θk ). Expanding now in terms of θk − θl , we get 1 g 2 (x − θk )g 2 (y − θl ) dx dy |x − y| 3 3 R R 1 g 2 (x)g 2 (y) dx dy = |x − y − (θk − θl )| R3 R3 1 2 2 2 g (x)g (y) − 4π θk − θl  dx dy g 4 (x) dx. ≈ |x − y| R3 R3 R3 The first part is of no relevance as it cancels with the normalization, so we just have to look at the Gaussian measure   T /η 2
4πλη 1 2 exp − e−λη|l−k| θk − θl  g 4 (x) dx . Z 2 3 R k,l=1

At the beginning, the path measure is tied down, so that we put θ1 ≈ 0, and consider that as a density for θ2 , . . . , θT /η . Then θT /η has covariance matrix 

−1 ∞
T 4πλη 2 ≈ g 4 (x) dx k 2 e−ληk I η R3 k=0 −1  ∞ T λ2 T 4π 4 2 −x  I. g (x) dx x e dx I = ≈ η λ 2 η R3 8π R3 g 4 (x) dx 0 Therefore, D (λ) =

 λ2 + o λ2 , 8π R3 g 4 (x) dx 

as λ → 0. This is of course very far from a mathematical proof. The Spohn-heuristics has however recently been verified in a much simpler case, namely for a onedimensional plane rotator model with a Kac-type interaction by Petermann [61]. Petermann does not use large deviation theory, but relies on the Griffiths inequalities which cannot be applied to the polaron, but the result gives some confidence that the result should be correct.

3. One-dimensional pinning-depinning transitions

I present in this chapter two results on one-dimensional random walks interacting with a layer. This layer, for the random walk, is just the path identical to zero. The interaction presented in the two sections are slightly different, but the effects are quite similar. The first section discusses what in physics literature is called a wetting transition. Here the layer is acting as a hard wall in the sense that the random walk has to stay on one side, but there is also an attractive interaction between the random walk and the wall. There is a considerable literature around such wetting-transitions, an I present here only the very most simple case where such a transition occurs. Of considerable interest are cases where the random walk is replaced by a random surface, and then, of course, similar questions for more complicated random interfaces, like interfaces in Ising type models, but I leave this out. In the second section I discuss a pinning-depinning transition for a model where the interaction with the “wall” is produced by random components of the random walk, which make the “nodes” of the random walk either be inclined to be on the positive side of the wall, or on the negative. This is a very simple model of a polymer chain whose components are either “oil repellent” or “water repellent”, and which is placed at an interface between water below and oil above. The components on the polymer chain are randomly placed, and one would like to know what the effect is.

3.1 Wetting transition for a one dimensional random walk We consider a standard random walk in one dimension (discrete time), i.e. the probability measure Pn on def

Ωn = {ω = (ω0 = 0, ω1 , . . . , ωn ) : |ωi − ωi−1 | = 1 for i = 1, . . . , n}, gives equal weight 2−n to all the paths. It will be convenient in this section, although it is not important at all, to work with a tied down walk. Therefore, let 0 def = {ω ∈ Ω2n : ω2n = 0} , Ω2n

106

3 One-dimensional pinning-depinning transitions

0 and again, P2n is the uniform distribution. Let β > 0 as usual be a positive coupling parameter. We first discuss only the case with an attraction to the “wall” (0, . . . , 0) . So we define

 2n−1
1 0 Pˆ2n,β (ω) = exp β 1ωi =0 P2n (ω). Z2n,β i=1

This is an extremely simple object, much simpler, of course, than the models discussed in Chapter 2. We prove that it localizes for every positive β > 0: Proposition 3.71. For any β > 0 a) def

 1 1 log Z2n,β = − log 1 − e−2β > 0. n→∞ 2n 2

f (β) = lim

b) There exist m(β) > 0, and A(β) such that ˆ2n,β (ωi ωj ) ≤ A(β) exp(−m(β)|i − j|). E Proof. a) is quite easy. Let q (l) , l ∈ 2N, be the distribution of the standard first return time to 0 : def

q (2l) = P2l (ωj = 0, 1 ≤ j ≤ 2l − 1, ω2l = 0) . The exact distribution is well known, and the generative function is 
z l q (l) = 1 − 1 − z 2 , z < 1. l

Furthermore



Z2n,β =

m≥1



9m

0=l0
j=1

q (lj − lj−1 )

P2n (ω2n = 0)

.

√ It is well known that the denominator is of order 1/√ n, so we only have to 42n,β . If 0 ≤ λ < 1 − e−2β , we get take care of the numerator, call it Z ∞
n=1

42n,β = λ2n Z

∞
m=1

eβ(m−1)


∞ n=1

q (2n) λ2n

m

√ 1 − 1 − λ2 √ = , 1 − eβ 1 − λ2

√ and this diverges for λ ↑ 1 − e−2β . From this, the exact form of f (β) follows. We next prove that a) implies b), which is very simple, too.

3.1 Wetting transition for a one dimensional random walk

107

Consider the random set Γ2n of “dry” points in {1, . . . , 2n − 1}, i.e. the points where the random walk visits 0. Clearly, if A = { 1 , . . . , m−1 }, 0 = 0 < 1 < . . . < m−1 < m = 2n, then P (A) =

m :

f ( j − j−1 ).

j=1 def We set [0,2n] (A) = Pˆ2n,β (Γ2n = A). Then β . [0,2n] (A) = P2n (Γ2n = A)eβ(m−1) /Z2n 0 and Pˆ2n,β have exactly the same law Conditionally on {Γ2n = A}, both P2n on paths just by choosing the excursions independently on all the intervals [ i−1 , i ]. If i ≥ j and A ∩ [i, j] = ∅, then ωi and ωj are independent under this conditioned law. We therefore get
0 ˆ2n,β (ωi ωj ) = [o,2n] (A)E2n (ωi ωj |Γ2n = A). (3.1) E A:A∩[i,j]=∅

If 0 ≤ k < ≤ 2n, we denote by Mk, the set of subsets A of {1, . . . , 2n − 1} such that A ∩ (k, ) = ∅, A ∪ {0, 2n} ⊃ {k, }.If A ∈ Mk, , k < i < j < , then 0 |E2n (ωi ωj |Γ2n = A)| ≤ − k.

(3.2)

Remark also that for A ∈ Mk, , B ⊂ (k, ) [0,n] (A ∪ B) =

[0,n] (A)[k,] (B)Z−k,β . f ( − k)

Therefore
A∈Mk,

[0,n] (A) =







[0,n] (A)[k,] (B) ≤

A∈Mk, B⊂(k,)

f ( − k) Z−k,β

Using this together with a) and (3.1) and (3.2) proves b). We now change the model in a way which leads to an interesting localization-delocalization transition similar to the one encountered in Section 1. This had been observed by M. Fisher [40] in his Boltzmann Lectures. We just replace the ordinary random walk by one having a “hard wall” condition, meaning that the walk has to stay positive. Therefore, we consider + 0 Ω2n = {ω ∈ Ω2n : ωi ≥ 0 ∀i},

108

3 One-dimensional pinning-depinning transitions

+ + + and P2n be the uniform distribution on Ω2n . The large n behavior of P2n is well known: After Brownian scaling, the path measure converges to that + (ωn ) is of a Brownian excursion on a fixed time interval. In particular, E2n √ of order n. Taking a coupling parameter β > 0 we again introduce an attractive “polymer-to-wall” interaction by defining   2n−1
1 + + Pˆ2n,β (ω) = +,β P2n (ω) exp β 1ωj =0  . Z2n j=1

One then has the following result Theorem 3.72. (M. Fisher [40]). There exists βcr > 0 such that 1. For β < βcr a) f + (β) = lim

1 n→∞ n

+ log Z2n,β =0

b) The path measure converges after Brownian scaling to that of Brownian excursion. 2) For β > βcr a) f + (β) > 0 b) There exist A(β), m(β) > 0 such that covPˆ + (ωi , ωj ) ≤ A(β) exp(−m(β)|i − j|) 2n,β

uniformly in n, i, j. Proof. I prove here only the existence of βcr such that f + (β) = 0 for β ≤ βcr and f + (β) > 0 for β > βcr . The exponential decay in 2(b) is more delicate than in Proposition 3.71. There is a recent paper [45] where these properties are proved in details, covering even the critical case β = βcr . f + (β) is evidently ≥ 0, convex in β ≥ 0, and satisfies f + (0) = 0. It is also + from very easy to see that f (β) > 0 for large enough β (just estimate Z2n,β below by taking the single path which hits 0 at all even times). Therefore, the only issue is to prove that f + (β) = 0 for small enough β > 0. To do this, remark first that 9m   −m β(m−1) m 0<1 <...<m−1 <2n 2 j=1 q( j − j−1 ) + . Z2n,β = + 0 P2n (Ω2n )P2n (ω2n = 0) + 0 It is well known that P2n (Ω2n ) ∼ 1/n, and therefore, we only have to take + . If 0 ≤ λ < 1, then one has now, care of the numerator, call it again Z˜2n,β  m ∞ ∞ ∞


2n ˜ + −m β(m−1) 2n λ Z = 2 e f (2n)λ 2n,β

n=1

m=1

n=1

 −1   1 eβ 2 2 = (1 − 1 − λ ) 1 − (1 − 1 − λ ) . 2 2

3.2 A heteropolymer near an interface

109

Evidently, if β < log 2, then this is finite for all λ < 1, and therefore f + (β) = 0. With this, I close this short discussion on wetting transitions. There are much more refined results. There are two recent papers [45] and [38], treating this in much more details. As remarked already, there also had been considerable work on such wetting transitions for higher-dimensional surfaces, instead of the one-dimensional “random string”. Proofs for such surfaces are generally much more delicate, and many questions which are relatively easy for stings become much more delicate for surfaces. I don’t want to enter this topic here, but refer to some of the relevant recent papers [16], [27], [31], [44], [17].

3.2 A heteropolymer near an interface We consider in this section an interesting interaction of the random walk (the “heteropolymer”) with the wall which is produced by a random environment which is given by a sequence of i.i.d. random variables σi = ±1, 1 ≤ i < ∞, which are also independent of the random walk. We denote the law of the σi by P, but we will allow that the σi are asymmetric: P(σi = 1) = (1 + h)/2, and we write then Ph . Remark that Eh (σi ) = h. We always assume that h ≥ 0, the other case being symmetric. Fixing an environment σ ∈ {−1, 1}N , we consider a transformed path measure on Ωn by  n


def −n P-n,β,σ (ω) = 2 exp β σi ∆i (ω) Zn,β,σ i=1 def

where β ≥ 0 is the usual coupling parameter, ∆i (ω) =sign(ωi ), (sign(0) = 0), and  n



−n Zn,β,σ = 2 exp β σi ∆i (ω) . ω∈Ωn

i=1

We are interested in the behavior of P-n,β,σ for large n which hold almost surely with respect to to Ph . This model has been introduced in the physics literature as a very simplified model for a so called heteropolymer. Think of (i, ωi ) as a (directed) polymer on Z2 , modelling a polymer chain whose one end (0, 0) is attached at a border of two “liquids”, the heavier one (say water) occupying (i, j) ∈ Z2 , j < 0, and the lighter one (say oil) the upper space (i, j), j > 0. We think now that our polymer chain (i, ωi ), i ≥ 0, is composed of molecules which are either water-repellent, meaning σi = 1, or oil repellent, i.e. σi = −1, but we allow that they do not appear in equal amounts, and we assume that they are randomly placed. The polymer then gets larger weight when the  “energy” − σi ∆i is low.

110

3 One-dimensional pinning-depinning transitions + +

+ −

−

+ oil −

+

+ +

+ : water repellant

−

water

− : oil repellant

Fig. 3.1.

The model has been introduced in the physics literature [41], [42], and a mathematically rigorous treatment started with the paper by Sinai [65], and then in [15], [66], [4] and most recently by Biskup [8]. The paper [15] treats actually a slightly different model, where the σi are given as h+σi0 , and where the σi0 are symmetric. This is technically slightly more convenient at some places. There are also recent papers in the physics literature, see e.g. [60]. The main question was to discuss the localization-delocalization behavior of the model. Evidently, the random walk (ωi ) has essentially two basic strategies to get its energy lowered. One is just to hang around the “water-oil” interface and switching to the right side as often as possible. This would mean that the polymer gets localized near the interface. It is however not evident that this should be the dominant effect even in the case h = 0, for arbitrary β > 0. However, this is exactly what Sinai has proved. If h > 0, i.e. when the water repellent nodes dominate, then the path could just stay on the upper half, getting dominant satisfaction cheaply, and it is not clear if this is better or worse than to act more sophisticated by hanging around the interface, and dipping from time to time into the watery side in order to please the now fewer oil repellent nodes . The basic result in [15] is that for any h > 0, there is a transition from a delocalized region, when β is small, where it is essentially not worthwhile for the path to chase after satisfaction for the minority of oil-repellent guys, and a large β region where this is better than to stay lazily in the oil all the time. I have to add here that we did not prove in [15] that the path measure really behaves in this way, but we worked instead purely with free energy considerations. This is the same as the evaluation in Chapter 2 of the rough leading order asymptotics which suggests but does not prove how the path measures really behaves. I hope it had become clear in chapter that to determine really the behavior of the path measure is usually then still a different story. Proposition 3.73. a) For any β ≥ 0, 0 ≤ h ≤ 1 Φ(β, h) = lim

1

n→∞ n

log Zn,β,σ

3.2 A heteropolymer near an interface

111

exists Ph -a.s. and in L1 (Ph ) and does not depend on σ. b) Φ is convex as a function of β c) βh ≤ Φ(β, h) ≤ β. d) Φ is jointly continuous in (β, h). Proof. a) is a standard application of the subadditive ergodic theorem, and I refer to [15]. Remark that from L1 -convergence, one has  n


σ i ∆i . Φ(β, h) = lim Eh log E exp β n→∞

i=1

b) This is the standard application using the H¨ older inequality 

n n

E exp λβ σi ∆i + (1 − λ)β σi ∆i  ≤E

i=1

exp β

n


i=1

λ  σi ∆i

E exp

n


i=1

1−λ σi ∆i

.

i=1

c) The upper bound is trivial, and the lower bound nearly so, but it is of crucial importance for what follows: 

  n

 n

E exp β σi ∆i ≥ E exp β σi ∆i ; ωi > 0, 1 ≤ i ≤ n) i=1

 = exp β

i=1 n




σi P (ωi > 0, 1 ≤ i ≤ n).

i=1

√ The second factor is well known to be of order 1/ n, and therefore plays no rˆ ole for the free energy. So the bound follows. n d) is fairly evident from the fact that | i=1 σi ∆i | ≤ n. If h1 < h2 , we couple σ 1 = (σi1 ) with law Ph1 , and σ 2 with law Ph2 in the usual way such that σi1 ≤ σi2 for all i. Then   n n n 


  1 2 σi ∆i − σ i ∆i  ≤ (σi2 − σi1 ),    i=1

i=1

i=1

and using that this is ≤ 2hn eventually, w.p. 1, the (Lipshitz) continuity follows. From the proof of the lower bound in c), it is plausible that the path stays localized if Φ(β, h) > βh. That this is actually the case had been proved for h = 0 by Sinai (who did not prove Φ(β, 0) > 0), and in the full generality where Φ(β, h) > βh by Biskup [8]. The case Φ(β, h) = βh means that there

112

3 One-dimensional pinning-depinning transitions

is nothing better the path can do than just essentially to stay positive, at least not on an exponential scale. It is natural to expect that at least in the interior of this region, this means that the path measure Pˆnβ,σ except perhaps for a few steps at the beginning has to concentrate on the “oily” paths. To be precise, we suspect that the following is true: Conjecture 3.74. Assume that (β, h) is such that for some ε > 0 (which may depend on (β, h)) one has Φ(β, h ) = βh for h ≥ h − ε. Then lim sup Pˆnβ,σ (max{i : ωi ≤ 0} ≥ K) = 0

K→∞ n

Ph - a.s. This would imply that the path, after Brownian rescaling has as limit distribution the Brownian meander Ph -a.s. The above conjecture looks very natural, there seem however to be considerable difficulties to prove it. For the rest of this section, we focus entirely on the behavior of Φ(β, h). Our first task is to prove that a “delocalized” phase, i.e. a region where Φ(β, h) = βh, really exists. As Φ(β, h) ≥ βh, this amounts to prove an upper bound for Φ. One of the basic tricks in the business of random media is to try to prove that the “quenched” free energy, i.e. our Φ(β, h) equals the “annealed” one, if for instance β is small. The annealed free energy is just lim

n→∞

1 log Eh Zn,β,σ n

1 Eh n→∞ n

which by Jensen dominates lim

(3.3)

log Zn,β,σ = Φ(β, h). It is however easily

checked that (3.3) is > βh for all β > 0, 0 ≤ h < 1. In fact  n


n Eh Zn,β,σ = cosh(β) E exp log (1 + h∆i tanh(β)) , i=1

and therefore lim

1

n→∞ n

log Eh Zn,β,σ = log(cosh(β) + h sinh(β)) > hβ

for h < 1. The last inequality is easily checked: The two sides agree at h = 1, and then the l.h.s. is concave and bigger than the r.h.s. at h = 0. Therefore, we are never able to prove Φ(β, h) = βh in this way except for h = 1, where it is trivial. However, a slight modification of the above argument leads to

3.2 A heteropolymer near an interface

113

Proposition 3.75. a) Φ(β, h) = βh if cosh (2β) − 1 . sinh (2β)

h≥

b) If (β, h) satisfies Φ(β, h) = βh, then Φ(β , h ) = β h if β > β, h > h, and   1 − h 2(β  −β) h − h −2β e + e ≤1 (3.4) 1−h 1−h Proof. a) The trick is to take a (trivial) part out of Z which is handled in a “quenched way”, and discuss the other part by an estimate of the type discussed above.  n


Zn,β,σ = E exp β σ i ∆i i=1

 = exp β

n






σi E exp β

i=1

n


σi (∆i − 1) .

i=1

Evidently, n1 log of the first factor goes to βh, so we have to estimate the second, which we do by Jensen.  n


Eh log E exp β σi (∆i − 1) i=1 n :

≤ log E

[cosh(β(∆i − 1)) + h sinh(β(∆i − 1))]

i=1

≤ log [(cosh(2β) − h sinh(2β)) ∨ 1] ≤ 0, if h ≥ (cosh (2β) − 1) / sinh (2β). b) This comes by a modification of the above argument. If h > h, we denote by (σi ) the signs distributed according to Ph , and then we choose, conditionally on (σi ) independent τi with τi = 2 w.p. (h − h)/(1 − h) if σi = −1, and τi = 0 otherwise. Then σi + τi is distributed according to Ph . We just write P for the joint law. Then if β > β 

n
E log E exp β (σi + τi )(∆i − 1) i=1

2



= E log E exp β  ×E

 exp

n


σi (∆i − 1)

i=1

n
i=1

 3   . (β − β)σi + β τi )(∆i − 1)  σ 



114

3 One-dimensional pinning-depinning transitions

 E

 exp

n
i=1

=

n :

    (β − β)σi + β τi )(∆i − 1)  σ 

exp [(β − β)σi (∆i − 1)]

i=1

/

× 1σi =1 + 1σi =−1



h − h 2β(∆i −1) 1 − h e + 1−h 1−h

0

≤1 for any choice of (σi ), (∆i ) as soon as (3.4) is satisfied. From this, b) follows. Theorem 3.76. For any β > 0 there exists hc (β) ∈ (0, 1) such that Φ(β, h) > 0 for h < hc (β), and Φ(β, h) = 0 for h ≥ hc (β). The function β → hc (β) has the following properties: a) β → hc (β) is continuous and non-decreasing. b) lim hc (β) = 1. β→∞

c) lim sup hc (β)/β ≤ 1. β↓0

d) lim inf hc (β)/β > 0. β↓0

Before I start with the proof, some comments. The largest part of [15] had actually been spent in proving that limβ↓0 hc (β)/β exists in (0, ∞), a fact which had been predicted in the physics literature, and “identifying” in a sense this limit. The proof of this is however rather delicate for reasons I will indicate below. I cannot give the details of this here. So I stick with proving just c), d). One might think that a discussion of the β → 0 limit leads to a perturbation expansion, but this is not the case. The tangent of hc (β) at β = 0 seems to be a complicated object and we have not much information about it except that it exists. Proof of Theorem 3.76. From Proposition 3.75, we see that for any β ≥ 0, if h is sufficiently close to 1, Φ(β, h) = βh. Remark also that  n


1 σi (∆i − 1) , Φ(β, h) − βh = lim Eh log E exp β n→∞ n i=1 and from this we see that Φ(β, h) − βh is non-increasing in h. As it is also continuous, we conclude that there exists hc (β) ∈ [0, 1) such that Φ(β, h) > βh if h < hc (β) and Φ(β, h) = βh if h ≥ hc (β). That β → hc (β) is nondecreasing follows from the convexity of Φ in β which implies (together with Φ(0, h) = 0) that if Φ(β, h) − βh > 0, then Φ(β , h) − β h > 0 for all β > β. The continuity of hc follows from Proposition 3.75 which implies that there are no upward jumps. Therefore, we have proved a).

3.2 A heteropolymer near an interface

115

c) is also a consequence of Proposition 3.75 a) and we are left with proving b) and d). Remark that d) in particular implies hc (β) > 0 for all β > 0. The proofs of b) and d) of course just require a lower bound for Φ. We follow a standard approach for proving lower bounds in large deviation theory, by changing the measure. Although our new measure is good enough to produce the lower bounds we are looking for, the reader will realize that (in contrast to the situation we had in Chapter 2) our reference measures are not the “correct” ones. The first observation is that the only relevant information needed about the random walk are the successive return times to 0: η0 = 0, ηj+1 = inf{i > ηj : ωi = 0}. Set ∆ηj = ηj − ηj−1 . The ∆ηj are i.i.d. random variables. As before, we put q( ) = P (∆ηj = l). The ηj define random points, or more formally random variables Zk , k ∈ N, where Zk = 1 if k = ηj for some j, and Zk = 0, otherwise. We define Fn = σ(Zk : k ≤ n). Our partition function has a trivial recasting in terms of the (Zk ) process by just integrating out the sign of the excursion between two successive returns to 0. Put ψ(x) = log cosh(x), and     ηj −1 τn n


ψ β σi  + ψ β σi  , Hn,β (Z, σ) = j=1

i=ηj−1 +1

i=ητn +1

where τn = max{k ≤ n : ηk ≤ n}. Then evidently Zn,β,σ = EnZ exp [Hn,β (Z, σ)] , where EnZ refers to taking the expectation for the sequence Z0 = 1, Z1 , . . . , Zn . Of course, we can introduce now also the Gibbs measure on the Z-sequence (for fixed σ) with this Hamiltonian. This however is a very complicated object, with a very complicated many body interaction (for fixed σ). Despite of this fact, we perform just a very simple change of measure, changing just the distribution f to q γ ( ) =

 1 q( )( 1 − γ 2 ) , 1−γ

γ > 0. We write PnZ,γ for the corresponding distribution of Z0 , . . . , Zn . Then by Jensen

 dPnZ,γ Z,γ Zn,β,σ = En exp Hn,β (Z, σ) − log (3.5) dPnZ ≥ exp(EnZ,γ Hn,β (Z, σ) − k(PnZ,γ |PnZ )), where k(µ|ν) is the usual Kullback-Leibler relative entropy. We now only have to choose the appropriate γ for getting the desired lower bounds. I stick to proving d) in Theorem 3.76, i.e. prove the bound for β ∼ 0. The β → ∞ is not difficult either, and I will leave this to the reader.

116

3 One-dimensional pinning-depinning transitions

Although this very crude argument is giving a lower bound proving that hc has a positive slope at 0, it seems clear that it could not produce the correct tangent (which we know to exist by considerations I will indicate below). First, we estimate the entropy  τn γ : qγ dPnZ,γ >n−ητn q ( )  (η = − η ) j j−1 dPnZ q >n−ητn q( ) j=1 ≤ (1 − γ)−τn −1 (1 − γ 2 )n/2 . Therefore k(PnZ,γ |PnZ ) ≤ − log(1 − γ)Enγ (τn + 1) +

n log(1 − γ 2 ). 2

By the optional sampling theorem, we get lim

1

n→∞ n

Enγ (τn + 1) =

γ , 1+γ

and therefore 1 1 γ log(1 − γ) + log(1 − γ 2 ). lim sup k(PnZ,γ |PnZ ) ≤ − 1+γ 2 n→∞ n

(3.6)

On the other hand, we have    ηj τn

1 1 lim inf EEnZ,γ Hn (Z, σ) ≥ lim inf EnZ,γ  Eψ β σi  n→∞ n n→∞ n j=1 i=ηj−1 +1  η  1
γ Z,γ = σi E Eψ β γ+1 n i=1 Now, we would ηj to the convex function ψ, but estimating ηj like to apply Jensen σj ) ≥ ψ(βE( j=1 σj )) = ψ(βη1 h) would kill the baby. In just Eψ(β( j=1 fact, this would mean that we just use that the σj have a average density of +1 of the right rate, and would not use the fluctuations. The fact that there is a positive slope of the hc -line is coming from the fluctuations of the σi . We use a slightly Jensen to the conditioned  more clever argument, by applying η1  η1 σj ≥ ηj h . Remark that P( j=1 σj ≥ η1 h) ≥ c1 (h) > 0, law P ·  j=1 where c(h) → 1/2 for h → 0. Therefore      η1 η1 η1


Eψ(β σj ) ≥ c1 (h)ψ βη1 h + βE  (σj − h)  (σj − h) ≥ 0   j=1 j=1 j=1 √ ≥ c1 (h)ψ(βη1 h + βc2 (h) η1 ),

3.2 A heteropolymer near an interface

117

where again, c2 (h) stays bounded away from 0 as h → 0. Combining this with (3.6) and (3.5), we get γc1 (h) Z,γ √ E ψ(βη1 h + βc2 (h) η1 ) 1+γ n γ 1 + log(1 − γ) − log(1 − γ 2 ) 1+γ 2 √ γc1 (h)
= q( )(1 − γ 2 )/2 ψ(βh + βc2 (h) ) 2 1−γ

Φ(β, h) ≥



γ 1 + log(1 − γ) − log(1 − γ 2 ). 1+γ 2 We now let β → 0, γ = bβ, h = aβ; a, b to be chosen later on. Then we get 1 Φ(β, βa) aβ 2  2  √ b b2 c1 b ∞ dx exp − x ψ(ax + c2 x) − , ≥ 3/2 a 0 x 2 2a

lim inf β→0

where we have used that q( ) ∼ c1 −3/2 , if → ∞ along even numbers. We can still choose b at our liking, so we take b = Ka, and let a → 0. Then   ∞ √ dx Ka2 x ψ(ax + c2 x) exp − lim inf 3/2 a→0 2 x 0 1 √ dx ≥ ψ(c2 x)dx > 0. 3/2 x 0 Therefore, as K is arbitrary lim inf lim inf a→0

β→0

1 Φ(β, βa) = ∞. aβ 2

In particular, lim inf β→0

1 Φ(β, βa) > 1 aβ 2

if a is small enough. This proves part d) of the Theorem. As remarked, I leave part b) to the reader. The β ∼ 0 case is actually somewhat puzzling at first sight. Let us look at the symmetric h = 0 case. The above argument for the lower bound shows also that Φ(β, 0) is of order β 2 . It is in fact true that limβ→0 β −2 Φ(β, 0) exists and is positive (see Proposition 3.79 below). Choosing γ in the above proof of order β gives a lower bound of the correct order β 2 (but not the correct constant). The procedure for this lower bound is somewhat naive. Our reference measure just shortens the excursions from 0 of the random

118

3 One-dimensional pinning-depinning transitions

walk by changing the distribution q. This happens completely regardless what the σ’s are. The σ’s play only a rˆ ole for choosing the signs of the excursion. This aspect is somewhat hidden in the proof above, as we have integrated out these signs immediately, but the effect can easily be reconstructed. If we have an excursion, of length , say, and S is the sum over the σ’s, then the excursion chooses the “oily side” with probability eβS /(eβS + e−βS ). Now, as remarked, it is certainly not the case that this measure on paths describes the true Gibbs measure accurately, but as the lower bound obtained using this measure is of the correct order, at least for β ∼ 0, we might guess that the true Gibbs measure has qualitatively about similar properties. For the correct Gibbs measure, the σ’s obviously influence the place where the excursions occur, and not just the sign as in the above strategy for the lower bound. For β → 0 the problem rescales to a problem on the Brownian motion. ole of the random walk, Let {ωt }t≥0 be a standard Brownian, playing the rˆ ole and {σt } an independent Brownian motion whose derivative plays the rˆ of the random environment. The model we would then naturally have is a transformation of the law of the first motion by {σt }, a coupling parameter β > 0, and a drift parameter h: 

T PˆT,β,h,σ (dω) = exp β sign(ωs )(dσ(s) + hds) PT (dω)/ZT,β,h,σ , 0

where PT (dω) is standard Wiener measure, and we define - h) = lim 1 log ZT,β,h,σ . Φ(β, T →∞ T - h) exists in (0, ∞) and has similar It is in fact not difficult to prove that Φ(β, - has a very properties as Φ introduced in Proposition 3.73. In contrast to Φ, Φ simple rescaling property which just comes from Brownian rescaling: Setting σ(t) = ˜ σ (t/2 ), we get 

T

ZT,β,h,σ = E exp β 

sign(ωt )(dσ(t) + hds) 0



= E exp β



T / 2

sign(ωs 2 )(d˜ σ (s) + hds) . 0

Remark now that if we put ω ˜ s = ωs 2 /, the sign is not influenced by the scaling, and therefore, we get - h) = 1 Φ(β, Φ(β, h). 2

(3.7)

3.2 A heteropolymer near an interface

119

- h), we therefore have If we put Λ(h) = Φ(1, - h) = β 2 Λ(h/β). Φ(β, It is not difficult to prove along the same lines as in the proof of Theorem 3.76 the following ˜ c > 0 such that Proposition 3.77. There is a critical value h ˜ c. a) Λ(h) = h if h ≥ h ˜ c. b) Λ(h) > h if h < h There seems to be no decent representation of hc for instance as some solution of a variational problem. Crude arguments like the ones before just show • • • •

Λ is continuous Λ(h) ≥ h for all h Λ(h) = h for h large enough Λ(0) > 0,

which imply the above proposition. It is however unclear if arguments of the type described before are able to characterize the correct value 4 hc . Given the value of 4 hc , the rescaling property shows that the phase separation line for the continuous model in the (β, h) plane is just a straight line. The following result is quite plausible but the proof is very delicate and cannot be given here. Theorem 3.78. The phase separation line hc (β) (for the discrete model) is ˜ c. differentiable at 0 with derivative h I close this section with some comments about the above result. ˜ c is quite universal and We expect (but did not prove) that this tangent h does not much depend on special properties of the discrete model. Evidently, changing for instance the law of the random walk or the environment would lead to different phase separation curves, but the tangent at 0 should not depend on specific properties of the model. We believe that there cannot be a particularly easy derivation of the theorem from standard invariance principles. There is an intermediate result which is easier to prove, namely convergence of the free energy. Proposition 3.79. lim

1

β→0 β 2

Φ(β, hβ) = Λ(h).

120

3 One-dimensional pinning-depinning transitions

The proof of this proposition considerably simpler than that of Theorem 3.78, but it is not easy either. The delicacy is that we do not really have good control of the true Gibbs measure. Essentially, one has to show that this Gibbs measures goes over, in the β → 0 limit to the corresponding Gibbs measure for the continuous model, which is as complicated (or even more so). Of course, we are not able to prove that on the level of Gibbs measures, but only on the level of free energies. Unfortunately, the proposition does not even quite imply the Theorem ˜ c , then Λ(h) > h, and therefore Φ(β, βh) > 3.78, but only half of it. If h < h 2 β h for small enough β. This shows that the slope of hc (β) at β = 0 is at ˜ c . However, if h > h ˜ c then the Proposition implies only that least h lim

1

β→0 β 2

Φ(β, hβ) = h.

This of course does not imply that Φ(β, hβ)/β 2 equals h for small enough β > 0, which is required for the proof of Theorem 3.78. The proof of the last fact uses repeated applications of much more complicated versions “semiannealed” trick, as in the proof of Proposition 3.75, but in only very implicitly defined situations. I think that the proof is of some methodological interest, but it is far too involved to be presented here.

References

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Part II

Edwin Perkins:

Dawson–Watanabe Superprocesses and Measure-valued Diffusions

E. Bolthausen, E. Perkins, A. van der Vaart: LNM 1781, P. Bernard (Ed.), pp. 125–329, 2002. c Springer-Verlag Berlin Heidelberg 2002


Table of Contents Page Glossary of Notation

129

I. Introduction

132

II. Branching Particle Systems and Dawson-Watanabe Superprocesses II.1. Feller’s Theorem II.2. Spatial Motion II.3. Branching Particle Systems II.4. Tightness II.5. The Martingale Problem II.6. Proof of Theorem II.5.11 II.7. Canonical Measures II.8. Historical Processes

135 137 143 148 159 172 178 187

III. Sample Path Properties of Superprocesses III.1. Historical Paths and a Modulus of Continuity III.2. Support Propagation and Super-L´evy Processes III.3. Hausdorff Measure Properties of the Supports III.4. One-dimensional Super-Brownian Motion and Stochastic PDE’s III.5. Polar Sets III.6. Disconnectedness of the Support III.7. The Support Process

193 200 207 215 223 240 244

IV. Interactive Drifts IV.1. Dawson’s Girsanov Theorem IV.2. A Singular Competing Species Model-Dimension One IV.3. Collision Local Time IV.4. A Singular Competing Species Model-Higher Dimensions

247 257 262 276

V. Spatial Interactions V.1. A Strong Equation V.2. Historical Brownian Motion V.3. Stochastic Integration on Brownian Trees V.4. Pathwise Existence and Uniqueness V.5. Martingale Problems and Interactions V.6. Appendix

281 283 292 300 310 314

References

318

Outline of Lectures at St. Flour Introduction Particle Sytems and Tightness (II.1-II.4) The Martingale Problem and Non-linear Equation (II.4-II.8) Path Properties of the Support of Super-Brownian Motion (III.1-III.3) Polar Sets (III.5-III.6) Interactive Drifts (IV) Spatial Interactions 1. Stochastic Integration on Trees and a Strong Equation (V.1-V.3) 8. Spatial Interactions 2. Pathwise Existence & Uniqueness, and the Historical Martingale Problem (V.4-V.5) 9. Interacting Particle Systems 1. The Voter Model 10. Interacting Particle Systems 2. The Contact Process

1. 2. 3. 4. 5. 6. 7.

Note. A working document with Ted Cox and Rick Durrett was distributed to provide background material for lectures 9 and 10.

Glossary of Notation |α|/N ≤ t < (|α| + 1)/N = ζ α , i.e., α labels a particle alive at time t Lebesgue measure of A the set of points less than a distance δ from the set A Aφ + gφ weak generator of path-valued Brownian motion–see Lemma V.2.1 see Proposition V.2.6 generator of space-time process–see prior to Proposition II.5.8 bp → bounded pointwise convergence bE the set of bounded E-measurable functions BSMP a cadlag strong Markov process with x → P x (A) Borel measurable for each measurable A in path space B(E) the Borel σ-field on E C C(Rd ) C(E) continuous E-valued functions on R+ with the topology of uniform convergence on compacts Cb (E) bounded continuous E-valued functions on R+ with the supnorm topology functions in Cb (Rd ) with bounded continuous partials of order k or less Cbk (Rd ) ∞ d functions in Cb (Rd ) with bounded continuous partials of any order Cb (R ) CK (E) continuous functions with compact support on E with the supnorm topology continuous functions on a locally compact E with a finite limit at ∞ C (E) C(g)(A) the g-capacity of a set A–see prior to Theorem III.5.2 C Borel σ-field for C = C(Rd ) Ct sub-σ-field of C generated by coordinate maps ys , s ≤ t α∼t |A| Aδ Ag φ Aˆ A¯τ,m  A

D

≡ D(E) Ds Df d

equal in distribution the space of cadlag paths from R+ to E with the Skorokhod J1 topology the set of paths in D(E) which are constant after time s smooth functions of finitely many coordinates on R+ × C –see Example V.2.8 D(n, d) space of Rn×d -valued integrands–see after Proposition V.3.1 ∆ cemetary state added to E as a discrete point D(A) domain of the weak generator A–see II.2 and Proposition II.2.2  T D(A) domain of weak space-time generator–see prior to Proposition II.5.7 ˆ D(A) domain of the weak generator for path-valued Brownian motion –see Lemma V.2.1 D(∆/2) domain of the weak generator of Brownian motion–see Example II.2.4 D the Borel σ-field on the Skorokhod space D(E) the canonical right-continuous filtration on D(E) (Dt )t≥0 e−W (φ) eφ (W ) E the Borel σ-field on E the non-negative E-measurable functions E+ ˆ E {(t, y(· ∧ t)) : y ∈ D(E), t ≥ 0} r β if β > 0, (log 1/r)−1 if β = 0 fβ (r) Fˆ F × B(C(Rd )) Ft × Ct Fˆt Fˆt∗ the universal completion of Fˆt

130 FX gβ (r) Gε φ G(f, t)

Superprocesses the Borel σ-field on ΩX r −β if β > 0, 1 + (log 1/r)+ , if β = 0 and 1, if β < 0 see (IV.3.4)
t sup Ps f (x)ds 0

G(X) h−m h(r) hd (r) Hts,y bp H H+ I

x

∪δ>0 cl{(t, x) : t ≥ δ, x ∈ S(Xt )}, the graph of X the Hausdorff h-measure–see Section III.3 L´evy’s modulus function (r log(1/r))1/2 r 2 log+ log+ 1/r if d ≥ 3, r 2 (log+ 1/r)(log+ log+ log+ 1/r) if d = 2 the Ht measure of {w : w = y on [0, s]}, s ≤ t, y(·) = y(· ∧ s) the bounded pointwise closure of H the set of non-negative functions in H ∞  N{0,...,n} = {(α0 , . . . , αn ) : αi ∈ N, n ∈ Z+ } n=0

IBSMP time inhomogeneous Borel strong Markov process–see after Lemma II.8.1 I(f, t) stochastic integral of f on a Brownian tree–see Proposition V.3.2 K the compact subsets of Rd Lip1 Lipschitz continuous functions with Lipschitz constant and supnorm ≤ 1 (LE) Laplace functional equation–see prior to Theorem II.5.11 local martingale problem for Dawson-Watanabe superprocess (LM P )ν with initial law ν–see prior to Theorem II.5.1 the collision local time of X = (X 1 , X 2 )–see prior to Remarks IV.3.1 Lt (X) (log x) ∨ ee log+ (x) LW (φ) the Laplace functional of the random measure W , i.e., E(e−W (φ) ) 2 Lloc see after Lemma II.5.2 space of probabilities on E with the topology of weak convergence M1 (E) MF (E) the space of finite measures on E with the topology of weak convergence MFt (D) the set of finite measures on D(E) supported by paths which are constant after time t the Borel σ-field on MF (E) MF Mloc the space of continuous (Ft )-local martingales starting at 0 (ME) mild form of the nonlinear equation–see prior to Theorem II.5.11 (M P )X0 martingale problem for Dawson-Watanabe superprocess with initial state X0 –see Proposition II.4.2 ˆ Ω Ω × C(Rd)      t ΩH [τ, ∞) H· ∈ C [τ, ∞), MF D(E) : Ht ∈ MF (D) ∀t ≥ τ ΩH ΩH [0, ∞) the space of continuous MF (E)-valued paths ΩX the space of cadlag MF (E)-valued paths ΩD pt (x) standard Brownian density pt (x − y) pxt (y) P the σ-field of (Ft )-predictable subsets of R+ × Ω 
t   Ptg φ(x) E x φ(Yt ) exp g(Ys ) ds PX0 Pν

0

the law of the DW superprocess on (ΩX , FX ) with initial state X0 –see Theorem II.5.1 the law of the DW superprocess with initial law ν

Superprocesses

131

ˆT P

the normalized Campbell measure associated with KT , i.e., ˆ T (A × B) = P(1A KT (B))/m(1) P (PC) x → P x is continuous (QLC) quasi-left continuity, i.e., Y is a Hunt process–see Section II.2 Qτ,m the law of the historical process starting at timeτ in state m–see Section II.8 S(Xt ), the range of X on I R(I) t∈I

R(I) R

R(I)is the  closed  range of X on I R [δ, ∞) is the range of X.

S(µ) St S (SE) t Tb

the closed support of a measure µ S(Xt ) simple P × E-measurable integrands–see after Lemma II.5.2 strong form of nonlinear equation–see prior to Theorem II.5.11 [N t]/N bounded (Ft )t≥τ -stopping times

δ>0

ucb

→ Uλ w ⇒ Wt y/s/w y t (·) ζα

convergence on E which is uniform on compacts and bounded on E the λ resolvent of a Markov process weak convergence of finite (usually probability) measures ˆ the coordinate maps on D(E) the path equaling y up to s and w(t − s) thereafter y(t ∧ ·) the lifetime of the αth branch–see after Remark II.3.2

I. Introduction Over the years I have heard a number of complaints about the impenetrable literature on measure-valued branching processes or Dawson-Watanabe superprocesses. These concerns have in part been addressed by some recent publications including Don Dawson’s St. Flour notes (Dawson (1993)), Eugene Dynkin’s monograph (Dynkin (1994)) and Jean-Francois Le Gall’s ETH Lecture Notes (Le Gall (1999)). Nonetheless, one still hears that several topics are only accessible to experts. However, each time I asked a colleague what topics they would like to see treated in these notes, I got a different suggestion. Although there are some other less flattering explanations, I would like to think the lack of a clear consensus is a reflection of the large number of different entry points to the subject. The FlemingViot processes, used to model genotype frequencies in population genetics, arise by conditioning the total mass of a superprocess to be one (Etheridge and March (1991)). When densities exist (as for super-Brownian motion in one spatial dimension) they typically are solutions of parabolic stochastic pde’s driven by a white noise and methods developed for their study often have application to large classes of stochastic pde’s (e.g. Mueller and Perkins (1992), Krylov (1997b), Mytnik (1998) and Section III.4). Dawson-Watanabe superprocesses arise as scaling limits of interacting particle systems (Cox, Durrett and Perkins (1999, 2000)) and of oriented percolation at criticality (recent work of van der Hofstad and Slade (2000)). Rescaled lattice trees above eight dimensions converge to the integral of the super-Brownian cluster conditioned to have mass one (Derbez and Slade (1998)). There are close connections with class of nonlinear pde’s and the interaction between these fields has led to results for both (Dynkin and Kuznetsov (1996,1998), Le Gall (1999) and Section III.5). They provide a rich source of exotic path properties and an interesting collection of random fractals which are amenable to detailed study (Perkins (1988), Perkins and Taylor (1998), and Chapter III). Those looking for an overview of all of these developments will not find them here. If you are looking for “the big picture” you should consult Dawson (1993) or Etheridge (2000). My goal in these notes is two-fold. The first is to give a largely self-contained graduate level course on what John Walsh would call “the worm’s-eye view of superprocesses”. The second is to present some of the topics and methods used in the study of interactive measure-valued models. Chapters II and III grew out of a set of notes I used in a one-semester graduate course on Superprocesses. A version of these notes, recorded by John Walsh in a legible and accurate hand, has found its way to parts of the community and in fact been referenced in a number of papers. Although I have updated parts of these notes I have not tried to introduce a good deal of the more modern machinery, notably Le Gall’s snake and Donnelly and Kurtz’s particle representation. In part this is pedagogical. I felt a direct manipulation of branching particle systems (as in II.3,II.4) allows one to quickly gain a good intuition for superprocesses, historical processes, their martingale problems and canonical measures. All of these topics are described in Chapter II. In the case of Le Gall’s snake, Le Gall (1999) gives an excellent and authoritative treatment. Chapter III takes a look at the qualitative properties of Dawson-Watanabe superprocesses. Aside from answering a number of natural questions, this allows us to demonstrate the effectiveness of the various tools used to study branching diffusions including the related nonlinear parabolic pde,

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133

historical processes, cluster representations and the martingale problem. Although many of the results presented here are definitive, a number of open problems and conjectures are stated. Most of the Exercises in these Chapters play a crucial role in the presentation and are highly recommended. My objective in Chapters II and III is to present the basic theory in a middling degree of generality. The researcher looking for a good reference may be disappointed that we are only considering finite variance branching mechanisms, finite initial conditions and Markov spatial motions with a semigroup acting on the space of bounded continuous functions on a Polish space E. The graduate student learning the subject or the instructor teaching a course, may be thankful for the same restrictions. I have included such appendages as location dependent branching and drifts as they motivate some of the interactions studied in Chapters IV and V. Aside from the survey in Section III.7, every effort has been made to provide complete proofs in Chapters II and III. The reader is assumed to have a good understanding of continuous parameter Markov processes and stochastic calculus–for example, the first five Chapters of Ethier and Kurtz (1986) provide ample background. Some of the general tightness results for stochastic processes are stated with references (notably Lemma II.4.5 and (II.4.10), (II.4.11)) but these are topics best dealt with in another course. Finally, although the Hausdorff measure and polar set results in Sections III.3 and III.5 are first stated in their most general forms, complete proofs are then given for slightly weaker versions. This means that at times when these results are used, the proofs may not be self-contained in the critical dimensions (e.g. in Theorem III.6.3 when d = 4). A topic which was included in the original notes but not here is the Fleming-Viot process (but see Exercise IV.1.2). The interplay between these two runs throughout Don Dawson’s St. Flour notes. The reader should really consult the article by Ethier and Kurtz (1993) to complete the course. The fact that we are able to give such a complete theory and description of Dawson-Watanabe superprocesses stems from the strong independence assumptions underlying the model which in turn produces a rather large tool kit for their study. Chapters IV and V study measure-valued processes which may have state-dependent drifts, spatial motions and branching rates (the latter is discussed only briefly). All of the techniques used to study ordinary superprocesses become invalid or must be substantially altered if such interactions are introduced into the model. This is an ongoing area of active research and the emphasis here is on introducing some approaches which are currently being used. In Chapter IV, a competing species model is used to motivate careful presentations of Dawson’s Girsanov theorem for interactive drifts and of the construction of collision local time for a class of measure-valued processes. In Chapter V, a strong equation driven by a historical Brownian motion is used to model state dependent spatial motions. Section IV.4 gives a discussion of the competing species models in higher dimensions and Section V.5 describes what is known about the martingale problems for these spatial interactions. The other sections in these chapters are again self-contained with complete arguments. There are no new results contained in these notes. Some of the results although stated and well-known are perhaps not actually proved in the literature (e.g. the disconnectedness results in III.6) and some of the proofs presented here are, I hope, cleaner and shorter. I noticed that some of the theorems were originally derived

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using nonstandard analysis and I have standardized the arguments (often using the historical process) to make them more accessible. This saddens me a bit as I feel the nonstandard view, clumsy as it is at times, is pedagogically superior and allows one to come up with novel insights. As one can see from the outline of the actual lectures, at St. Flour some time was spent on rescaled limits of the voter model and the contact process, but these topics have not made it into these notes. A copy of some notes prepared with Ted Cox and Rick Durrett on this subject was distributed at St. Flour and is available from me (or them) upon request. We were trying to unify and extend these results. As new applications are still emerging, I decided it would be better to wait until they find a more definitive form than rush and include them here. Those who have seen earlier versions of these notes will know that I also had planned to include a detailed presentation of the particle representations of Donnelly and Kurtz (1999). In this case I have no real excuse for not including them aside from running out of time and a desire to keep the total number of pages under control. They certainly are one of the most important techniques available for treating interactive measure-valued models and hence should have been included in the second part of these notes. There a number of people to thank. First the organizers and audience of the 1999 St. Flour Summer School in Probability for an enjoyable and at times exhausting 2 12 weeks. A number of suggestions and corrections from the participants has improved these notes. The Fields Institute invited me to present a shortened and dry run of these lectures in February and March, and the audience tolerated some experiments which were not entirely successful. Thanks especially to Siva Athreya, Eric Derbez, Min Kang, George Skoulakis, Dean Slonowsky, Vrontos Spyros, Hanno Treial and Xiaowen Zhou. Most of my own contributions to the subject have been joint and a sincere thanks goes to my co-authors who have contributed to the results presented at St. Flour and who have made the subject so enjoyable for me: Martin Barlow, Ted Cox, Don Dawson, Rick Durrett, Steve Evans, Jean-Francois Le Gall and Carl Mueller. Finally a special thanks to Don Dawson and John Walsh who introduced me to the subject and have provided ideas which can be seen throughout these notes.

II. Branching Particle Systems and Dawson-Watanabe Superprocesses 1. Feller’s Theorem Let {Xik : i ∈ N, k ∈ Z+ } be i.i.d. Z+ -valued random variables with mean 1 and variance γ > 0. We think of Xik as the number of offspring of the ith individual Zk in the k th generation, so that Zk+1 = Xik (set Z0 ≡ 1) is the size of the k + 1st i=1

generation of a Galton-Watson branching process with offspring distribution L(Xik ), the law of Xik . w We write an ∼ bn iff lim an /bn = 1 and let ⇒ denote weak convergence of n→∞

finite (usually probability) measures. Theorem II.1.1. (a) (Kolmogorov (1938)) P (Zn > 0) ∼ 2/nγ as n → ∞.  w (b) (Yaglom (1947)) P Znn ∈ · | Zn > 0 ⇒Z, where Z is exponential with mean γ/2. Proof. (a) This is a calculus exercise in generating functions but it will be used on several occasions and so we provide a proof. Let fn (t) = E(tZn ) for t ∈ [0, 1], and f (t) = f1 (t) (here 00 = 1 as usual). A simple induction shows that fn is the n-fold composition of f with itself. Then Dominated Convergence shows that f  and f  are continuous on [0, 1], where the appropriate one-sided derivatives are taken at the endpoints. Moreover f  (1) = E(Xik ) = 1 and f  (1) = var(Xik ) = γ. As f is increasing and strictly positive at 0 (the latter because Xik has mean 1 and is not constant), we must have 0 < f (0) ≤ fn (0) ↑ L ≤ 1 and f (L) = L. Note that f  (t) = E(Z1 tZ1 −1 ) < 1 for t < 1 and so the Mean Value Theorem implies that f (1) − f (t) < 1 − t and therefore, f (t) > t, for t < 1. This proves that L = 1 (as you probably already know from the a.s. extinction of the critical branching process). Set xn = fn (0) and (II.1.1)

yn = n(1 − xn ) = nP (Zn > 0).

A second order Taylor expansion shows that 1 − xn+1 = f (1) − f (xn ) = 1 − xn − Therefore

f  (zn ) (1 − xn )2 for some zn ∈ [xn , 1]. 2

yn+1 = (n + 1)(1 − xn+1 ) f  (zn ) = (n + 1)[1 − xn − (1 − xn )2 ] 2

y y 2 f  (zn )  n − n2 = (n + 1) n n 2

f  (zn )  1 1 − yn (1 + n−1 ) . = yn 1 + n 2

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Now let γ1 < γ < γ2 and δ > 0. Note that limn→∞ xn = 1 and limn→∞ f  (zn ) = γ, and so we may choose n0 so that for n ≥ n0 , (1 − xn )γ2 /2 < δ,

(II.1.2) (II.1.3)



  1 1 yn 1 + (1 − yn γ2 /2) ≤ yn+1 ≤ yn 1 + (1 − yn γ1 /2) , n n

and therefore, (II.1.4)

yn+1 > yn if yn < 2/γ2 , and yn+1 < yn if yn > 2/γ1 .

Claim yn > γ22 (1 − δ) eventually. Note first that if n1 ≥ n0 satisfies yn1 ≤ γ22 (1 − δ/2) (there is nothing to prove if no such n1 exists), then the lower   bound in (II.1.3) shows that yn1 +1 ≥ yn1 1 + 2nδ 1 . Iterating this bound, we see that there is an n2 > n1 for which yn2 > γ22 (1 − δ/2). Now let n3 be the first n > n2 for which yn ≤ γ22 (1 − δ) (again we are done if no such n exists). Then yn3 −1 > γ22 (1 − δ) ≥ yn3 and so (II.1.4) implies that yn3 −1 ≥ γ22 . Therefore (II.1.3) shows that 

1 yn3 ≥ yn3 −1 1 + (1 − yn3 −1 γ2 /2) n3 − 1 yn3 −1 γ2  2

1− ≥ γ2 n3 − 1 2

2 (1 − xn3 −1 )γ2  1− = γ2 2 2 > (1 − δ), γ2 the last by (II.1.2). This contradicts the choice of n3 and hence proves the required inequality for n ≥ n2 . A similar argument shows that yn ≤ γ21 (1 − δ) eventually. We thus have shown that limn→∞ yn = 2/γ and hence are done by (II.1.1). (b) will be a simple consequence of Theorem II.7.2 below, the proof of which will use (a). See also Section II.10 of Harris (1963). These results suggest we consider a sequence of critical Galton-Watson branch(n) (n) ing processes {Z0 : n ∈ N} as above but with initial conditions Z0 satisfying (n) (n) (n) Z0 /n → x, and define Xt = Z[nt] /n. Indeed it is an easy exercise to see from (n)

the above that X1 masses.

converges weakly to a Poisson sum of independent exponential

Notation. E denotes a Polish space. Let D(E) = D(R+ , E) be the Polish space of cadlag paths from R+ to E with the Skorokhod J1 -topology. Let C(E) = C(R+ , E) be the Polish space of continuous E-valued paths with the topology of uniform convergence on compacts. Let Yt (y) = y(t) for y ∈ D(E).

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137

w

Theorem II.1.2. (Feller (1939, 1951)) X (n) ⇒X in D(R), where X is the unique solution of t γXs dBs , Xt = x +

(F E)

0

where B is a one-dimensional Brownian motion. Proof. We will prove much more below in Theorem II.5.1. The uniqueness holds by Yamada-Watanabe (1971). We call the above process Feller’s branching diffusion with parameter γ. 2. Spatial Motion We now give our branching population a spatial structure. Individuals are “located” at a point in a Polish space E. This structure will also usually allow us to trace back the genealogy of individuals in the population. Notation. E = Borel subsets of E ≡ B(E), Cb (E) = {f : E → R : f bounded and continuous} with the supnorm,  , D = B D(E) , (Dt )t≥0 is the canonical right-continuous filtration on D(E),
µ(f ) = f dµ for a measure µ and integrable function f . Assume (II.2.1)

Y = (D, D, Dt, Yt , P x ) is a Borel strong Markov process (BSMP)

with semigroup Pt f (x) = P x (f (Yt )). “Borel” means x → P x (A) is E-measurable for all A ∈ D. The other required properties here are Y0 = x P x −a.s. and the strong Markov property. Evidently our BSMP’s have cadlag paths. These assumptions are either much too restrictive or far too abstract, depending on your upbringing. At the risk of offending one of these groups we impose an additional condition: Pt : Cb (E) → Cb (E).

(II.2.2)

This is only needed to facilitate our construction of Dawson-Watanabe superprocesses as limits of branching particle systems and keep fine topologies and Ray compactifications at bay. Standard arguments (see Exercise II.2.1 below or the proof of Theorem I.9.21 in Sharpe (1988)) show that (II.2.2) implies Y is a Hunt process, i.e., if {Tn } are {Dt }-stopping times such that (QLC) Tn ↑ T < ∞ P x a.s., then Y (Tn ) → Y (T ) P x − a.s. In particular, Yt = Yt− P x − a.s. for all t > 0. Definition. φ ∈ D(A) iff φ ∈ Cb (E) and for some ψ ∈ Cb (E), t φ(Yt ) − φ(Y0 ) −

ψ(Ys )ds is a P x -martingale for all x in E. 0

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It is easy to see ψ is unique if it exists and so we write ψ = Aφ for φ ∈ D(A). bp

Notation. → denotes bounded pointwise convergence. bp →ψ as t ↓ 0 for some Proposition II.2.1. φ ∈ D(A) ⇔ φ ∈ Cb (E) and Pt φ−φ t ψ ∈ Cb (E). In this case, ψ = Aφ and for any s ≥ 0, Ps φ ∈ D(A) and APs φ = Ps Aφ =

(II.2.3)

∂ Ps φ. ∂s

Proof. (⇐) If s ≥ 0, our assumption and the semigroup property show that Ps+t φ − Ps φ bp →Ps ψ t

as t ↓ 0 ∀s ≥ 0.

(QLC) implies Ps ψ(x) is continuous in s for each x. An easy calculus exercise shows that a continuous function with a continuous right derivative is differentiable, and so from the above we have ∂ Ps φ = Ps ψ, ∂s

(II.2.4) and so



P φ(Yt ) − φ(Y0 ) −

t

x

 ψ(Ys )ds = Pt φ(x) − φ(x) −

0

t Ps ψ(x)ds = 0. 0

The Markov property now shows the process in the above expectation is a martingale and so φ ∈ D(A) with Aφ = ψ. (⇒) Let φ ∈ D(A) and s ≥ 0. 

  t bp Pt+s φ(x) − Ps φ(x) /t = P x P Ys  Aφ(Yr )dr/t →Ps (Aφ)(x) as 



t ↓ 0,

0

where the last limit holds by Dominated Convergence. Taking s = 0, one completes the proof of (⇒). For s > 0, one may use the above argument and (⇐) to see Ps φ ∈ D(A) and get the first equality in (II.2.3). The second equality follows from (II.2.4) with ψ = Aφ. 
∞ −λt  Let Uλ f (x) = E x e f (Yt )dt denote the λ-resolvent of Y for λ > 0. 0

Clearly Uλ : Cb (E) → Cb (E) by (II.2.2). Proposition II.2.2. (a) ∀φ ∈ D(A) Uλ (λ − A)φ = φ. Uλ φ ∈ D(A) and (b) ∀φ ∈ Cb (E)

(λ − A)Uλ φ = φ.

Superprocesses Proof. (a)

∞ Uλ Aφ(x) =

139

e−λt Pt Aφ(x)dt

0

∞ =

e−λt

 ∂ Pt φ(x) dt ∂t

(by II.2.3)

0

∞ t=∞  + λ e−λt Pt φ(x)dt = e−λt Pt φ(x) t=0

0

= −φ(x) + λUλ φ(x). (b) Uλ φ(Yt ) = E

x

 ∞

   e−λu φ(Yu )duDt eλt

t

(II.2.5)

t 

= e Mt − e−λu φ(Yu )du , λt

0

where Mt denotes the martingale E x


∞

 e−λu φ(Yu )du | Dt . Some stochastic calcu-

0

m

lus shows that (= means equal up to martingales)   t s t t m λs  −λu  Uλ φ(Yt )= λe φ(Yu )du ds− φ(Ys )ds = λUλ φ(Ys )−φ(Ys )ds, Ms − e 0

0

0

0

where we have used (II.2.5). This implies Uλ φ ∈ D(A) and AUλ φ = λUλ φ − φ. Notation. bE (respectively, E+ ) is the set of bounded (respectively, non-negative) bp E-measurable functions. If H ⊂ bE, H is the smallest set containing H and closed bp under →, and H+ is the set of non-negative functions in H. bp

Corollary II.2.3. D(A)

= bE,

bp

(D(A)+ )

= bE+ .

bp

bp

Proof. If φ ∈ Cb (E), Pt φ→φ as t ↓ 0 and so it follows that λUλ φ→φ as λ → ∞, bp

and so φ ∈ D(A) . The result follows trivially. Exercise II.2.1. Prove that Y satisfies (QLC). Hint. (Following the proof of Theorem I.9.21 of Sharpe (1988).) Let X = lim Y (Tn ) ∈ {Y (T −), Y (T )}. It suffices to consider T bounded and show E x (g(X)h(YT )) = E x (g(X)h(X)) for all g, h ∈ Cb (E) (why?). As in the proof of Corollary II.2.3 it suffices to consider h = Uλ f , where f ∈ Cb (E) and λ > 0. Complete the required argument by using the strong Markov property of Y and the continuity of Uλ f .

140

Superprocesses Here are some possible choices of Y .

Examples II.2.4. (a) Yt ∈ Rd is d-dimensional Brownian motion. Cb2 (Rd ) = {φ : Rd → R, φ is C 2 with bounded partials of order 2 or less} ⊂ D(A) 2 d o’s Lemma. In this case we will write D(∆/2) for and A = ∆φ 2 for φ ∈ Cb (R ) by Itˆ D(A). symmetric stable process of index α ∈ (0, 2) (b) Yt ∈ Rd is the  d-dimensional  α scaled so that P x eiθ·Yt = eiθ·x−t|θ| , where |y| is the Euclidean norm of y. If ν(dy) = c|y|−d−α dy for an appropriate c > 0, then for φ ∈ Cb2 (Rd ) ⊂ D(A)   y  Aφ(x) = φ(x + y) − φ(x) − ∇φ(x) · ν(dy), 1 + |y|2

as can be easily seen, e.g., from the stochastic calculus for point processes in IkedaWatanabe (1981, p. 65–67) (see also Revuz-Yor (1990, p. 263)). In both the above examples Cb∞ (Rd ) = {φ ∈ Cb (Rd ) : all partial derivatives of φ are in Cb (Rd )} is a core for A in that the bp-closure of {(φ, Aφ) : φ ∈ Cb∞ (Rd )} contains {(φ, Aφ) : φ ∈ D(A)}. To see this first note that if φ ∈ D(A) has compact support, then Pt φ(x) = pt (y − x)φ(y)dy ∈ Cb∞ (Rd ) for t > 0 because Y has a transition density, pt , all of whose spatial derivatives are bounded and continuous. In the stable case the latter is clear from Fourier inversion because α m |θ| pt (y − x)eiθ·y dy = |θ|m eiθ·x−t|θ| is bounded and integrable in θ for all m ∈ N. Now choose {ψn } ⊂ Cb∞ with compact support so that ψn ↑ 1 and {|x| ≤ n} ⊂ {ψn = 1} ⊂ {ψn > 0} ⊂ {|x| < n + 1}. If φ ∈ D(A) and φn = φψn , then an integration by parts shows that φn ∈ D(A) and Aφn = ψn Aφ + φAψn . The above shows that P1/n φn ∈ Cb∞ . Dominated bp

Convergence implies that P1/n φn →φ, and (II.2.3) and a short calculation shows that bp AP1/n φn = P1/n Aφn = P1/n (ψn Aφ + φAψn )→Aφ. This proves the above claim. Notation. M1 (E) is the space of probabilities on a Polish space E and its Borel σ-field, equipped with the topology of weak convergence. CK (E) is the space of continuous function on E with compact support, equipped with the topology of uniform convergence. Exercise II.2.2. Assume Yt ∈ Rd is d-dimensional Brownian motion with d > 2 and U0 f is defined as above but with λ = 0.

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141

(a) Show that if f
≥ 0 is Borel measurable on Rd , then as λ ↓ 0, Uλ f (x) ↑ U0 f (x) = kd |y − x|2−d f (y)ds ≤ ∞, for some kd > 0. (b) Show that U0 : CK (Rd ) → D(A), AU0 φ = −φ for all φ ∈ CK (Rd ), and U0 Aφ = −φ for all φ ∈ CK (Rd ) ∩ D(A). bp

Hint. One approach to (b) is to show that for φ ∈ CK (Rd ), as λ ↓ 0, Uλ φ→U0 φ ∈ Cb bp

and AUλ φ→ − φ. Example II.2.4. (c) Suppose, in addition, that our BSMP Y satisfies   (P C) x → P x is continuous from E to M1 D(E) . This clearly implies (II.2.2). Under (PC) we claim that the previous  hypotheses are satisfied by the pathvalued process t → (t, Y t ) ≡ t, Y (· ∧ t) ∈ R+ × D(E). To be more precise, ˆ = {(t, y t) : t ≥ 0, y ∈ D(E)} with the subspace topology it inherits from let E ˆ and if y, w ∈ D(E) and s ≥ 0, let R+ × D(E), Eˆ = B(E),   y(t) t < s  (y/s/w)(t) = ∈ D(E) . w(t − s) t ≥ s ˆ is Polish as it is a closed subset of the Polish space R+ × D(E). Note that E ˆ →E ˆ denote the coordinate maps and for (s, y) ∈ E, ˆ Definition. Let Wt : D(E) ˆ ˆ ˆ define Ps,y on D(E) with its Borel σ-field, D, by   Pˆs,y (W· ∈ A) = P y(s) (s + ·, y/s/Y · ) ∈ A , i.e., under Pˆs,y we run y up to time s and then tag on a copy of Y starting at y(s). Proposition II.2.5. (W, (Pˆs,y )(s,y)∈Eˆ ) is a BSMP with semigroup ˆ → Cb (E). ˆ Pˆt : Cb (E) ˆ t ) be the canonical Proof. This is a routine if somewhat tedious exercise. Let (D ˆ Fix u ≥ 0 and to check the Markov property right-continuous filtration on D(E). ˆ and ˆ A∈D ˆ u , ψ be a bounded measurable function on E at time u, let (s, y) ∈ E, T ≥ u. Also set A˜ = {w ∈ D(E) : (v → (s + v, (y/s/wv )) ∈ A} ∈ Du , and

˜ ψ(w) = ψ(s + u + t, y/s/(wu+t)), w ∈ D(E).

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Then ˜ u+t )dP y(s) ˆ 1A ψ(Wu+t )dPs,y = 1A˜ (Y )ψ(Y ˜ u (ω)/u/Y t )dP y(s) (ω) (by the Markov property for Y ) = 1A˜ (Y (ω))P Yu(ω) (ψ(Y = 1A˜ (Y (ω))P Yu(ω) (ψ(s + u + t, (y/s/Y u (ω))/s + u/Y t )) dP y(s) (ω) = 1A (s + ·, y/s/Y · (ω))Pˆs+u,y/s/Y u (ω) (ψ(Wt ))dP y(s) (ω) = 1A PˆWu (ω) (ψ(Wt ))dPˆs,y (ω). This proves the Markov property at time u. ˆ and suppose (sn , yn ) → Turning now to the semigroup Pˆt , let f ∈ Cb (E) ˆ Note that if T > sup sn is a continuity point of y∞ , then (s∞ , y∞ ) in E. n yn (sn ) = yn (T ) → y∞ (T ) = y∞ (s∞ ). Therefore by (PC) and Skorohod’s representation (see Theorem 3.1.8 of Ethier and Kurtz (1986)) we may construct a sequence of processes, {Yn : n ∈ N ∪ {∞}} such ˆ a.s. Now use the fact that that Yn has law P yn (sn ) and lim Yn = Y∞ in D(E) sn → s∞ , (yn , Yn ) → (y∞ , Y∞ ) in D(E)2 , and Yn (0) = yn (sn ) imply

yn /sn /Yn → y∞ /s∞ /Y∞ as n → ∞ in D(E).

This is a standard exercise in the Skorokhod topology on D(E) which is best left to the reader. Note that the only issue is the convergence near s∞ and here the condition Yn (0) = yn (sn ) avoids the possibility of having distinct jump times approach t ) a.s. s∞ in the limit. The above implies that limn→∞ f (yn /sn /Ynt ) = f (y∞ /s∞ /Y∞ and therefore t )) = Pˆt f (s∞ , y∞ ). lim Pˆt f (sn , yn ) = lim E(f (yn/sn /Ynt )) = E(f (y∞ /s∞ /Y∞

n→∞

n→∞

ˆ → Cb (E). ˆ The strong Markov property of W now This shows that Pˆt : Cb (E) follows from this and the ordinary Markov property by a standard approximation of a stopping time by a sequence of countably-valued stopping times. Also the Borel measurability of (s, y) → Pˆs,y (ψ(W )) is now clear for ψ a bounded continuous function of finitely many coordinates, in which case this function is continuous by the above, and hence for all bounded and measurable ψ by the usual bootstrapping argument. Finally Pˆs,y (W (0) = (s, y)) = 1 is clear from the definitions. Exercise II.2.3. Let Yt be a Feller process with a strongly continuous semigroup Pt : C (E) → C (E), where C (E) is the space of continuous functions on a locally compact metric space (E, d) with finite limit at ∞. Show that (PC) holds. Hint. Let xn → x. It suffices to show {P xn } is tight on D(E) (why?). By Aldous’ criterion (Walsh (1986), Thm. 6.8(a)) it suffices to consider a sequence of stopping

Superprocesses times {Tn }, Tn ≤ t0 < ∞, a sequence δn ↓ 0, and show   lim P xn d(YTn +δn , YTn ) > ε = 0 n→∞

143

∀ε > 0.

Exercise II.2.4. Give an example of a Markov process satisfying the hypotheses (II.2.1) and (II.2.2) of this Section but for which (PC) fails. Hint. Take E = R+ and consider the generator Af (y) = (f (1)−f (0))1(y = 0)+(f (1/y)−f (y))1(0 < y ≤ 1)+y(f (1)−f (y))1(y > 1). Thanks go to Tom Kurtz for suggesting this example. 3. Branching Particle Systems Let Y be the E-valued Markov process from the previous Section and introduce a drift function g ∈ Cb (E) and branching variance (or rate) function γ ∈ Cb (E)+. Recall that G+ denotes the non-negative elements in the set G. The state space for our limiting and approximating processes will be MF (E), the space of finite measures on E with the topology of weak convergence, and we choose an initial state X0 ∈ MF (E). For N ∈ N and x ∈ E, ν N (x, ·) ∈ M1 (Z+ ) is the offspring law for a parent ucb

located at x. We assume x → ν N (x, ·) is measurable and satisfies ( → denotes convergence on E which is uniform on compacts and bounded on E): gN (x) ucb where gN →g as N → ∞, (a) kν N (x, dk) = 1 + N   ucb (II.3.1) (b) Var ν N (x, ·) = γN (x) where γN →γ as N → ∞, (c) ∃ δ > 0 such that sup k 2+δ ν N (x, dk) < ∞. N,x

Remarks II.3.1. (1) At the cost of complicating our arguments somewhat, (II.3.1) (c) could be weakened to the uniform integrability of k 2 with respect to {ν N (x, ·) : x ∈ E, N ∈ N}. (2) Given γ ∈ Cb (E)+ and g ∈ Cb (E) it is not hard to see there is a sequence {ν N } satisfying (II.3.1). In fact there is a k ∈ Z, k ≥ 2 and functions pN and qN such that ν N (x, ·) = δ0 (1 − pN (x) − qN (x)) + δ1 pN (x) + δk qN (x) will satisfy (II.3.1) with gN ≡ g for N ≥ N0 . Exercise II.3.1. Prove this. Hint. A simple calculation shows that if (II.3.1(a,b)) hold for ν N as above with gN = g, then pN (x) = 1 + g(x)/N − αN (x)/(k − 1) and qN (x) = αN (x)/(k 2 − k), where αN (x) = γN (x) + g(x)/N + g(x)2 /N 2 . √ Let ηN = g∞ /N + g2∞ /N 2 + g∞ / N and set γN (x) = γ(x) ∨ ηN . Show that you can choose k sufficiently large so that pN (x), qN (x) ≥ 0, pN (x) + qN (x) ≤ 1 for N large, and (II.3.1) is valid with gN = g for such an N .

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We now describe a system of branching particles which undergo near critical branching at times k/N according to the laws νN (x, ·) where x is the location of the parent. In between branch times particles migrate as independent copies of the process Y from the previous section. It will be important to have a labeling scheme to refer to the branches of the resulting tree of Y -processes. We follow the arboreal labeling of Walsh (1986) – in fact this section is really the missing Ch. 10 of Walsh’s SPDE notes. We have decided to work in a discrete time setting but could just as well work in the continuous time setting in which inter-branch intervals are exponential rate N random variables. We label particles by multi-indices α∈I=

∞ 

N{0,...,n} = {(α0 , . . . , αn ) : αi ∈ N, n ∈ Z+ }.

n=0

Let |(α0 , . . . , αn )| = n be the generation of α and write β < α ⇔ β = (α0 , . . . , αi ) ≡ α|i for some i ≤ |α|, i.e. if β is an ancestor of α. We let α ∨ k = (α0 , . . . αn , k) denote the k th offspring of α and α ∧ β denote the “greatest common ancestor” of α and β (set α ∧ β = φ if α0 = β0 and |φ| = −∞), and let πα = (α0 , . . . , αn−1 ) denote the parent of α if n > 0. Adjoin ∆ to E as a discrete point to form E∆ = E ∪ {∆} and let P ∆ be point N × D(E∆ )N × ZN mass at the constant path identical to ∆. Let Ω = E∆ + and let F denote its product σ-field. Sample points in Ω are denoted by ω = ((xi , i ∈ N), (Y α , α ∈ I), (N α , α ∈ I)). Now fix N ∈ N and define a probability P = P N on (Ω, F ) as follows: (II.3.2) (a) (xi , i ≤ MN ) is a Poisson point process with intensity N X0 (·) and xi = ∆ if i > MN . (b) Given Gn = σ(xi , i ∈ N) ∨ σ(N β , Y β , |β| < n), {Y α : |α| = n} are (conditionally) independent and (for |α| = n)      n + 1   n  ∈ AY · ∧ = Y·πα (ω) P (Y α ∈ A|Gn )(ω) = P xα0 (ω) Y · ∧ N N   |α| where Y πα ≡ xα0 if |α| = 0. That is, Y α · ∧ = Y πα (·)and given Gn , N {Y α |[|α|/N,(|α|+1)/N] : |α| = n} evolve as independent copies of Y starting from Y πα (|α|/N ), and stopped at (|α| + 1)/N. (c) Given G¯n = Gn ∨ σ(Y α : |α| = n), {N α : |α| = n} are (conditionally) independent and P (N α ∈ ·|G¯n )(ω) = ν N (Y α ((|α| + 1)/N, ω), ·). It should be clear from the above that {Y α : α ∈ I} is an infinite tree of branching Y processes, where Ytα = Ytβ for 0 ≤ t < (|α ∧ β| + 1)/N . Let t = [N t]/N for t ≥ 0 where [x] is the integer part of x, set T = TN = {kN −1 : k ∈ Z+ } and

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145

let τ = 1/N . It will be convenient to work with respect to the right continuous filtration given by      N α α α σ (Ys : |α| = N t, s ≤ r) Ft = Ft = σ (xi )i∈N , (Y , N )|α|t

It also will be useful to introduce the slightly larger σ-field   F t = Ft ∨ σ Y α : |α| = N t . Here are some consequences of our definition of P for each α ∈ I and t = |α|/N : (II.3.3) {(Y α , Fs ) : s ≥ t} is a Markov process,and for all s ∈ [t, t + τ ], α P (Y α (s + ·) ∈ A|Fs )(ω) = P Ys (ω) (Y (· ∧ (t + τ − s)) ∈ A) a.s. for all A ∈ D. (II.3.4) P (Y α ∈ A|F0 )(ω) = P xα0 (ω) (Y (· ∧ (t + τ )) ∈ A) a.s. for all A ∈ D. α , ·) a.s. P (N α ∈ ·|F¯t ) = ν N (Yt+τ

(II.3.5)

Clearly (II.3.5) is a restatement of (II.3.2)(c). (II.3.4) should be clear from (II.3.3) and (II.3.2)(b) (one can, for example, induct on |α|). To prove (II.3.3), it suffices to prove the stated equality for s ∈ [t, t + τ ), so fix such an s. The stated result is an easy consequence of (II.3.2)(b) if Fs is replaced by the smaller σ-field α α GNt ∨ FsY , where FsY is the right continuous filtration generated by Y α . Now use the fact that Hα ≡ σ(Y β : |β| = N t, β = α) is conditionally independent of Y α given GNt (by (II.3.2(b)) to see that the stated result is valid if Fs is replaced by α the larger σ-field GNt ∨ FsY ∨ Hα . Now condition this last equality with respect to Fs to obtain (II.3.3). M Remark II.3.2. If X0N = N1 1 N δxi , an easy consequence of (II.3.2)(a) and the weak law of large numbers is (II.3.6)

P

X0N (φ)→X0 (φ) as N → ∞ for any bounded measurable φ on E. P

Note also that E(X0N (·)) = X0 (·). From (II.3.6) it is easy to show that X0N →X0 . For example, one could use the existence of a countable convergence determining class of functions on E (see the proof of Theorem 3.4.4 in Ethier and Kurtz (1986)). Instead of assuming {xN i , i ≤ MN } is as in (II.3.2)(a), we could assume more , i ≤ M } are random points (MN is also random) chosen so generally that {xN N i M P N 2 N that X0N = N1 1 N δxN →X 0 , supN E(X0 (1) ) < ∞, and E(X0 (·)) ≤ c0 X0 (·) as i measures. The only required change is to include c0 as multiplicative factor in the upper bound in Lemma II.3.3 below. For example, we could assume {xi , i ≤ MN } are i.i.d. with law X0 (·)/X0 (1) and MN = [N X0 (1)] and set c0 = 1. The interpretation of (II.3.2) (b,c) is that for |α| = N t, the individual labelled by α follows the trajectory Y α on [t, t + τ ] and at time t + τ dies and is replaced

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by its N α children. The next step is to use the N α to prune the infinite tree of branching Y α processes. The termination time of the αth branch is  0, if xα0 = ∆      min (i + 1)/N : i < |α|, N α|i < αi+1 , if this set is not ∅ and xα0 = ∆ ζα =    (|α| + 1)/N, otherwise. Note in the first case, the αth particle was never born since α0 > MN . In the second case, αi+1 > N α|i means the αi+1 st offspring of α|i doesn’t exist. Finally in the last instance, the family tree of α is still alive at (|α| + 1)/N but we have run out of data to describe its state beyond this final time. ω We write α∼t (or α ∼ t) iff |α|/N ≤ t < (|α| + 1)/N = ζ α , i.e., iff α labels a particle alive at time t. Clearly α ∼ t iff α ∼ t. Note that we associate α with the particle alive on [|α|/N, (|α| + 1)/N ], although of course Ysα , s < (|α| + 1)/N describes the past history of its ancestors. From Feller’s theorem (Theorem II.1.2) it is natural to assign mass N1 to each particle alive at time t and define XtN =

1  1  δYtα = δY α , N α∼t N α∼t t

i.e., XtN (A) = #{Ytα ∈ A : α ∼ t}/N,

A ∈ E.

Since N α < ∞ for all α ∈ I a.s., clearly XtN ∈ MF (E) for all t ≥ 0 a.s. Note δYtα also that Y α ∈ D(E) for all α with xα0 = ∆ a.s., and therefore XtN = N1 α∼t   has sample paths in D MF (E) a.s. on each [t, t + τ ), and hence on all of R+ . The associated historical process is HtN =

  1  α ∈ MF D(E) . δY·∧t N α∼t

  Again H·N ∈ D R+ , MF (D(E)) . Therefore XtN is the (normalized) empirical measure of the particles alive at time t while HtN records the past trajectories of the ancestors of particles alive at time t. Clearly we have XtN (φ) = φ(yt )HtN (dy). Exercise II.3.2. Show that (i) {α ∼ t} ∈ Ft . (ii) XtN is Ft -measurable. (A trivial exercise designed only to convince you that |α| < N t in the above definition is correct.)   w Our goal is to show X N ⇒X in D MF (E) and characterize X as the unique solution of a martingale problem. The weak convergence of H N to the associated historical process H will then follow easily by considering the special case in Example II.2.4 (c). As any student of Ethier and Kurtz (1986) knows, the proof proceeds in two steps: 1. Tightness of {X N } and derivation of limiting martingale problem. 2. Uniqueness of solutions to the martingale problem.

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147

These are carried out in the next 3 sections. We close this section with a simple bound for the first moments.
Notation. P µ = P x (·)µ(dx), µ ∈ MF (E). If y ∈ C(E), let y t (s) = y(t ∧ s). Lemma II.3.3. Let g∞ = sup gN ∞ and g = sup{gN (x) : x ∈ E, N ∈ N}. N

(a) If ψ : D(R+ , E) → R+ is Borel, then for any t ≥ 0       E HtN (ψ) ≤ egt E X0 ψ(Y t ) ≤ eg∞ t E X0 ψ(Y t ) .     In particular, E XtN (φ) ≤ egt E X0 φ(Yt ) ∀φ ∈ E+ . (b) For all x, K > 0 and for all N ≥ N0 (g∞ ),   P sup XtN (1) ≥ x ≤ e3g∞ K X0 (1)x−1 . t≤K

  Proof. (a) Let Ps,t (ψ)(y) = E y(0) ψ(Y t ) | Y s = y , s ≤ t. We prove the result for t ≤ t by induction on t. If t = 0, then one has equality in the above. Assume the result for t ≤ t. Then  1   α N (ψ) = ψ Y·∧(t+τ ) N α , Ht+τ N α∼t and so  E

N (ψ) Ht+τ



      1 α α =E ψ Y·∧(t+τ ) E N | F t  N α∼t         1 α α ψ Y·∧(t+τ 1 + gN Yt+τ τ  =E (by (II.3.5)) ) N α∼t      1 α  (by (II.3.3)) Pt,t+τ ψ Y·∧t ≤ (1 + gτ )E  N α∼t   ≤ egτ egt E X0 Pt,t+τ ψ(Y·∧t) (by induction hypothesis)   g(τ +t) X0 ψ(Y·∧(t+τ ) ) . E =e 

Finally it should be clear from  above that the result follows for t ∈ (t, t + τ ).  the (b) Claim e2g∞ t XtN (1) is an F t -submartingale for N ≥ N0 (g∞ ). From the above calculation we have   N e2g∞ (t+τ ) E Xt+τ (1) | F t ≥ e2g∞ (t+τ ) (1 − g∞ τ )XtN (1) ≥ e2g∞ t XtN (1), for N ≥ N0 (g∞ ). The weak L1 inequality for non-negative submartingales and (a) now complete the proof. Remark II.3.4. It is clear from the above argument that if gN ≡ 0, then equality holds in (a).

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4. Tightness We first specialize Theorem 3.1 of Jakubowski (1986), which gives necessary and sufficient conditions for tightness in D(R+ , S), to the case S = MF (E). As E is Polish, MF (E) is also Polish–see Theorem 3.1.7 of Ethier and Kurtz (1986) for the corresponding result for M1 (E) from which the result follows easily. (An explicit complete metric is defined prior to Lemma II.7.5.) Therefore D(MF (E)) is also Polish and Prohorov’s theorem implies that a collection of laws on D(MF (E)) is tight iff it is relatively compact. Definition. A collection of processes {X α : α ∈ I} with paths in D(S) is Crelatively compact in D(S) iff it is relatively compact in D(S) and all weak limit points are a.s. continuous. Definition. D0 ⊂ Cb (E) is separating iff for any µ, ν ∈ MF (E), µ(φ) = ν(φ) ∀φ ∈ D0 implies µ = ν. of Theorem II.4.1. Let D0 be a separating class in Cb (E) containing 1. A sequence  cadlag MF (E)-valued processes {X N , N ∈ N} is C-relatively compact in D MF (E) iff the following conditions hold: (i) ∀ε, T > 0 there is a compact set KT,ε in E such that  c sup P sup XtN (KT,ε ) > ε < ε. N

t≤T

(ii) ∀φ ∈ D0 , {X·N (φ) : N ∈ N} is C-relatively compact in D(R+ , R). If, in addition, D0 is closed under addition, then the above equivalence holds when ordinary relatively compactness in D replaces C-relative compactness in both the hypothesis and conclusion. Remark. A version of this result is already implicit in Kurtz (1975) (see the Remark after Theorem 4.20). A proof of the sufficiency of the above conditions in the C-relatively compact setting is given at the end of this Section. All the ideas of the proof may be found in Theorem 3.9.1 and Corollary 3.9.2 of Ethier and Kurtz (1986). Although the C-relatively compact version is the result we will need, a few words are in order about the result in the general setting. (i) essentially reduces the result to the case when E is compact. In this case it is not hard to see there is a countable subset D0 ⊂ D0 closed under addition such that ψ(µ) = (µ(f ))f ∈D0  is a homeomorphism from MF (E) onto its image in RD0 . The same is true of    the map Xt → (Xt (φ))φ∈D from D MF (E) onto its image in D(R+ , RD0 ). To 0





complete the proof we must show D(R+ , RD0 ) and D(R+ , R)D0 are homeomorphic. This is the step which requires D0 to be closed under addition. As any scholar of the J1 -topology knows, X n → X and Y n → Y in D(R+ , R) need not imply (X n , Y n ) → (X, Y ) in D(R+ , R2 ), but it does if in addition X n + Y n → X + Y . See Jakubowski (1986) for the details. ! σ(Xs : s ≤ u) denotes the right-continuous filtration generNotation. FtX = u>t

ated by a process X. Let Af φ = (A + f )φ for φ ∈ D(A), f ∈ Cb (E).

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149

We will use Theorem II.4.1 to prove the following tightness result. Recall that our standing hypotheses (II.2.1), (II.2.2) and (II.3.1) are in force.   Proposition II.4.2. {X N } is C-relatively compact in D MF (E) . Each weak limit point, X, satisfies t ∀φ ∈ D(A), Mt (φ) = Xt (φ) − X0 (φ) −

Xs (Ag φ)ds is a continuous 0

(M P )X0 

 FtX -martingale such that M0 (φ) = 0 and M (φ)t =

t Xs (γφ2 )ds. 0

Proof. We take D0 = D(A) in Theorem II.4.1, which is a separating class by Corollary II.2.3. We first check (ii) in II.4.1. Let φ ∈ D(A) and define t Mtα

=

φ(Ytα )

−

φ(Ytα )

−

# " Aφ(Ysα )ds, t ∈ t, t + N −1 , t = |α|/N, α ∈ I.

t

Note that if we group the population at s + N −1 according to their parents at s we get  1   α N φ Ys+N −1 N α . Xs+N −1 (φ) = N α∼s Therefore, (II.4.1)

   1   α φ Ys+N −1 N α − φ Ysα N α∼s 

        1  α α α N α − 1 + gN Ys+N N −1 + N −2 φ Ys+N φgN Ys+N = −1 −1 −1 N α∼s α∼s

N N Xs+N −1 (φ) − Xs (φ) =

1  α M + −1 + N α∼s s+N

−1 s+N

s

1  Aφ (Ysα ) ds N α∼s

and XtN (φ) − XtN (φ) = (II.4.2)

1  (φ (Ytα ) − φ(Ytα )) N α∼t 1  α 1  Mt + Aφ (Ysα ) ds. N α∼t N α∼t t

=

t

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Sum (II.4.1) over s < t and then add (II.4.2) to arrive at    1   α α −1 XtN (φ) = X0N (φ) + φ Ys+N −1 N α − 1 + gN (Ys+N −1 )N N s
XsN (φgN )ds + N −2



   α φgN Ys+N − φgN (Ysα ) −1

s
0



 t   1 1 α α + Ms+N −1 + Mt  + XsN (Aφ)ds, N s
and therefore t (M P )

N

XtN (φ)

=X0N (φ)

+

Mtb,N (φ)

XsN (φgN )ds + δtN (φ)

+ 0

t + Mts,N (φ) +

XsN (Aφ)ds. 0

In the last line, the terms Mtb,N (φ), δtN (φ) and Mts,N (φ) are defined to be the corresponding terms in the previous expression. We start by handling the error term δ N . L1

Lemma II.4.3. sup |δtN (φ)|→0 as N → ∞ ∀K ∈ N. t≤K   Proof. If hN (y) ≡ E y |gN φ(YN −1 ) − gN φ(Y0 )| then          1  α α  E sup |δtN (φ)| ≤ N −1 E  E gN φ(Ys+N −1 ) − gN φ(Ys ) | Fs N α∼s t≤K s
K =

  E XsN (hN ) ds

(by (II.3.3))

0

K ≤ eg∞ K

  E X0 hN (Ys ) ds

0

= eg∞ K E X0



K 0

(Lemma II.3.3(a))

 |gN φ(Ys+N −1 ) − gN φ(Ys )| ds .

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Now since {Ys , Ys− : s ≤ K} is a.s. compact, (II.3.1)(a) and Dominated Convergence show that for some ηN → 0,   K   E sup |δtN (φ)| ≤ eg∞ K E X0 |gφ(Ys+N −1 ) − gφ(Ys )| ds + ηN t≤K

0

→ eg∞ K E X0



K

|gφ(Ys ) − gφ(Ys− )| ds



as N → ∞

0

= 0. Use (II.3.3) and argue as in the proof of Proposition II.2.1 to see that (Mtα , Ft )t∈[t,t+N −1 ] is a martingale. This and the fact that {α ∼ t} ∈ Ft (recall Exercise II.3.2) easily imply that (Mts,N (φ), Ft )t≥0 is a martingale. Lemma II.3.3 (a) with φ ≡ 1 implies integrability. Perhaps somewhat surprisingly, we now show that M s,N (φ) will not contribute to the limit as N → ∞. This is essentially a Strong Law effect. A moment’s thought shows that the fact that Y has no fixed time discontinuities (by (QLC)) must play an implicit role in the proof as the following result fails without it. L2

Lemma II.4.4. sup |Mts,N (φ)|→0 as N → ∞ ∀K > 0. t≤K  2 Proof. Let hN (y) = E y [φ(Y1/N ) − φ(Y0 )] . The definition of Mtα and an easy orthogonality argument shows that for K ∈ N         α 2  E E (Ms+N E M s,N (φ)2 = N −2 −1 ) | Fs K

s
≤ 2N −2



α∼s

 

   2 E  hN (Ysα ) + Aφ∞ N −2 

(by (II.3.3))

α∼s

s
 K ≤ 2E  XsN (hN ) + Aφ2∞ N −2 XsN (1)ds 0

(



≤ 2eg∞ K E X0 0

K

 2 φ(Ys+N −1 ) − φ(Ys ) ds



) + KN −2 Aφ2∞ X0 (1) .

In the last line we have used Lemma II.3.3(a) and argued as in the proof of Lemma II.4.3. As in that result the above expression approaches 0 as N → ∞ by Dominated Convergence. This proves the above expectation goes to 0 and the result follows by the strong L2 inequality for martingales.

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Recall that T = TN = {j/N : j ∈ Z+ }. We claim (Mtb,N (φ), F t )t∈T is also a martingale. To see this note that   b,N b,N (φ) | F t E Mt+N −1 (φ) − Mt      1   α α −1 | Ft φ Yt+N −1 E N α − 1 + gN (Yt+N = −1 )N N α∼t =0 by our definition of the conditional law of N α . Integrability of this increment is again clear from Lemma II.3.3(a) and E(|N α |) ≤ C. In view of the fact that our spatial martingales vanish in the limit we expect that the martingale Mt (φ) in (M P )X0 must arise from the “branching martingales” M b,N (φ). To analyze M b,N we use the following “well-known” result.  N Lemma II.4.5. Let MtN , F t t∈T be martingales with M0N = 0. Let N    N N N 2 M N t = E (Ms+N | F s ), and extend M·N and M N · to R+ as −1 − Ms 0≤s
right-continuous step functions. (a) If {M N · : N ∈ N} is C-relatively compact in D(R) and sup |M N (t + N −1 ) − M N (t)|→0 as P

(II.4.3)

N → ∞ ∀K > 0,

0≤t≤K

then {M N } is C-relatively compact in D(R). (b) If, in addition, + * 2 MtN + M N t : N ∈ N is uniformly integrable ∀t ∈ T, (II.4.4) w

then M Nk ⇒M in D(R) implies M is a continuous L2 martingale and w (M Nk , M Nk )⇒(M, M ) in D(R)2 . (c) Under (II.4.4), the converse to (a) holds. Proof. (a) is immediate from Theorems VI.4.13 and VI.3.26 of Jacod-Shiryaev (1987). A nonstandard proof (and statement) of the entire result may also be found in Theorems 8.5 and 6.7 of Hoover-Perkins (1983). (b) remains valid without the M N t term in (II.4.4) but the proof with this condition becomes completely elementary as the reader can easily check. The key ingredient in the above result is a predictable square function inequality of Burkholder (1973): (P SF ) ∃c = c(c0 ) such that if φ : R+ → R+ is continuous, increasing, φ(0) = 0 and φ(2λ) ≤ c0 φ(λ) for all λ ≥ 0, (Mn , Fn ) is a martingale, Mn∗ = supk≤n |Mk |, n    E (Mk − Mk−1 )2 | Fk−1 + E(M02 ) and d∗n = max1≤k≤n |Mk − Mk−1 |, M n = k=1



   1/2 + φ(d∗n ) . then E (φ(Mn∗ )) ≤ c E φ M n

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To apply the above Lemma to Mtb,N (φ), first note that by (II.3.2)(c), if α ∼ s, β ∼ s, and α = β, then N α and N β are conditionally independent given F¯s . The resulting orthogonality then shows that  2 b,N b,N E (φ) | F s Ms+N −1 (φ) − Ms  2 1  α 2 α −1 φ(Ys+N N α − (1 + gN (Ys+N ) | Fs = 2 −1 ) E −1 )N N α∼s =

1  α 2 α φ(Ys+N −1 ) γN (Ys+N −1 ). N 2 α∼s

Sum over s < t to see that t (II.4.5)

M

b,N



−2 εN t (φ) = N

XsN (1)ds, 0

0

where

t

XsN (φ2 γN )ds + εN t (φ) ≤ cφ

(φ)t =



α 2 α φ2 γN (Ys+N −1 ) − φ γN (Ys ).

s
Just as in Lemma II.4.3, for any K > 0, L1

sup |εN t (φ)|→0 as

(II.4.6)

N → ∞.

t≤K

We also see from the above that for 0 ≤ s < t ≤ K, M b,N (φ)t − M b,N (φ)s ≤ c sup XsN (1)|t − s|, s≤K

which in view of Lemma II.3.3(b) and Arzela-Ascoli implies the C-relative compactness of {M b,N (φ)· : N ∈ N} in D(R). To verify (II.4.3) and (II.4.4) we will use the following result, whose   proof we defer. Lemma II.4.6. sup E N

sup XtN (1)2

t≤K

< ∞ for any K > 0.

Exercise II.4.1 (a) If δ is as in (II.3.1(c)) use (PSF) to show    2+δ lim E  |∆M b,N (φ)(t)|  = 0 ∀K > 0. N→∞

t≤K

Hint: Conditional on F t , ∆M b,N (φ)(t) is a sum of mean 0 independent r.v.’s to which one may apply (PSF). (b) Use (a), (PSF), and Lemma II.4.6 to show that ∀K > 0   sup E N

2+δ

sup |Mtb,N (φ)|

t≤K

<∞

154

Superprocesses   2 and, in particular, sup |Mtb,N (φ)| : N ∈ N is uniformly integrable. t≤K   The uniform integrability of M b,N (φ)t : N ∈ N is clear from (II.4.5) and Lemma II.4.6, and so the above Exercise allows us to apply the relative compactness lemma for martingales (Lemma II.4.5) to conclude: (M b ) {M·b,N (φ) : N ∈ N} are C-relatively compact in D, and all limit points are w

continuous L2 martingales. If M·b,Nk (φ)⇒M· (φ), then w  b,N M· k (φ), M b,Nk (φ)· ⇒ (M· (φ), M (φ)) in D2 . Proof of Lemma II.4.6. Let φ = 1 in (M P )N and combine the terms t XsN (gN )ds + δtN (1) to see that (recall that g∞ = supN gN ) 0

t XtN (1)

(II.4.7)

≤

X0N (1)

+

Mtb,N (1)

XsN (1)ds.

+ g∞ 0

2

Doob’s strong L inequality and Lemma II.3.3(a) imply     b,N 2 E sup Mt (1) ≤ cE M b,N (1)K t≤K

 K ≤ cE  XsN (1)ds

(II.4.8)

(by (II.4.5))

0

≤ c(K)X0 (1). Jensen’s inequality and (II.4.7) show that for u ≤ K u sup XtN (1)2 t≤u

≤

cX0N (1)2

+ c sup t≤K

Mtb,N (1)2

+

XsN (1)2 ds.

2 cg∞ K 0

Consider the first u ≤ K at which the mean of the left side is infinite. If such a u exists, the last integral  has finite mean and so does the right-hand side. This contradiction shows E sup XtN (1)2 < ∞ and a simple Gronwall argument now t≤K

gives a bound uniform in N , namely for all u ≤ K,     N 2 2 N 2 E sup Xt (1) ≤ c sup E X0 (1) + c(K)X0 (1) ecg∞ Ku . t≤u

N

To complete the verification of (ii) in Theorem II.4.1, return to (M P )N . Recall P from Remark II.3.2 that X0N (φ)→X0 (φ) as N → ∞. The Arzela-Ascoli Theorem
t
t and Lemma II.3.3(b) show that XsN (φgN )ds and XsN (Aφ)ds are C-relatively 0

0

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155

compact sequences in D(R). Therefore (M P )N , (M b ) and Lemmas II.4.3 and II.4.4 show that {X·N (φ)} is C-relatively compact in D(R) for each φ in D(A), and (ii) is verified. We now give the Proof of the Compact Containment Condition (i) in Theorem II.4.1. Let ε, T > 0 and η = η(ε, T ) > 0 (η will be chosen below). As any probability on a Polish space is tight, we may choose a compact set K0 ⊂ D(E) so that P X0 (Y ∈ K0c ) < η. Let K = {yt , yt− : t ≤ T, y ∈ K0 }. It is easy to see K is compact in E (note that if tn → t and yn → y, then ynk (tnk ) → y(t) or y(t−) for some subsequence {nk } and similarly for ynk (tnk −)). Clearly P X0 (Yt ∈ K c Let

or Yt− ∈ K c

∃t ≤ T ) < η.

RtN = e2g∞ t HtN {y : y(s) ∈ K c for some 1  = e2g∞ t sup 1K c (Ysα ). N α∼t s≤t

s ≤ t}.

Claim RtN is an FtN -submartingale for N ≥ N0 . As R·N is increasing on [t, t + τ ), it suffices to show that   N N | Ft− ≥ 0 a.s. (II.4.9) E RtN − Rt− We have RtN

−

N Rt−

1 = N ≥ =

1 N 1 N +

 α∼t−N −1



α∼t−N −1



2g∞ t

e

sup 1K c (Ysα )N α s≤t

2g∞ (t−N −1 )

−e

sup 1K c (Ysα ) s
 −1 e2g∞ t N α − e2g∞ (t−N ) sup 1K c (Ysα ) s


e2g∞ t N α − gN (Ytα )/N − 1 sup 1K c (Ysα )

α∼t−N −1

1 N

)

(



α∼t−N −1

s
 e2g∞ t gN (Ytα )/N + 1 − e−2g∞ /N sup 1K c (Ysα ). s
N is 0. The second The conditional expectation of the first term with respect to Ft− term is at least

  1 e2g∞ t −g∞ /N + 1 − e−2g∞ /N sup 1K c (Ysα ) ≥ 0, for N ≥ N0 (g∞ ), N s
and (II.4.9) is proved. Now use the weak L1 inequality for submartingales and

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Lemma II.3.3 (a) to see that for N ≥ N0 (g∞ )   P sup XtN (K c ) > ε ≤ P sup RtN > ε t≤T

t≤T

−1

≤ε

E(RTN )

≤ e2g∞ T ε−1 egT P X0 (Ys ∈ K c ∃s ≤ T ) <ε by an appropriate choice of η = η(ε, T ). It is trivial to enlarge K if necessary to accommodate N < N0 (g∞ ), e.g. by using the converse of Theorem II.4.1 and the fact that each PN is trivially tight. Completion of Proof of Proposition II.4.2. By Theorem II.4.1 {X·N } is C relatively compact in D MF (E) . To complete the proof of Proposition II.4.2 we w must verify that all limit points satisfy (M P )X0 . Assume X Nk ⇒X. By a theorem of Skorohod (see Theorem 3.1.8 of Ethier-Kurtz (1986)) we may assume X Nk , X are defined on a common probability space and X Nk → X in D MF (E) a.s. Let
t
t φ ∈ D(A). Note that X → Xr (Aφ)dr and X → Xr (gφ)dr are continuous maps 0 0   from D R+ , MF (E) to C(R+ , R), and X → X· (φ) is a continuous function from D(MF (E)) to D(R). This, Lemmas II.4.3 and II.4.4, the convergence in probability of X0Nk (φ) to X0 (φ) (by (II.3.6)), the uniform convergence on compacts of gNk to g and condition (i) in Theorem II.4.1 allow us to take limits in (M P )Nk and conclude that M·b,Nk (φ) → M· (φ) in D(R+ , R) in probability where t Xt (φ) = X0 (φ) + Mt (φ) +

Xs (Aφ + gφ)ds. 0

By (M b ), M· (φ) is a continuous square integrable martingale. In addition, (M b ) ucb

together with (II.4.5), (II.4.6), γNk →γ and the compact containment condition (i) allow one to conclude t M (φ)t = P − lim

t XsNk (φ2 γNk )ds

k→∞ 0

Xs (φ2 γ)ds.

= 0

To complete the derivation of (M P )X0 we must show that Mt (φ) is an (FtX )martingale and not just a martingale with respect to its own filtration. To see this let s1 ≤ s2 . . . ≤ sn ≤ s < t, ψ : MF (E)n → R be bounded and continuous and use Exercise II.4.1 (b) to take limits in     E Mtb,Nk (φ) − Msb,Nk (φ) ψ XsN1k , . . . , XsNnk = 0. This shows that X satisfies (M P )X0 and thus completes the proof of Proposition II.4.2.

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Proof of Theorem II.4.1. We only prove the sufficiency of the two conditions as the necessity is quite easy. Let d be a complete metric on MF (E) and if x ∈ D(MF (E)) and δ, T > 0, set w(x, δ, T ) = sup{d(x(t), x(s)) : s, t ≤ T, |s − t| ≤ δ}. A standard result for general Polish state spaces states that {X N : N ∈ N} is C-relatively compact if and only if (II.4.10)

0 ⊂ MF (E) such that ∀ε, T > 0 there is a compact set Kε,T 0 / Kε,T for some t ≤ T ) ≤ ε, sup P (X N (t) ∈ N

and (II.4.11) ∀ε, T > 0 there is a δ > 0 so that lim sup P (w(X N , δ, T ) ≥ ε) ≤ ε. N→∞

For example, this follows from Corollary 3.7.4, Remark 3.7.3, and Theorem 3.10.2 of Ethier and Kurtz (1986). We first verify the compact containment condition (II.4.10). Let ε, T > 0. By condition (i) there are compact subsets Km of E so that c ) > 2−m ) < ε2−m−1 . sup P (sup XtN (Km N

t≤T

Take φ = 1 in condition (ii) and use (II.4.10) for the real-valued processes X·N (1) to see there is an R = R(ε, T ) so that sup P (sup XtN (1) > R) < ε/2. N

t≤T

Define c ) ≤ 2−m for all m ∈ N, and µ(1) ≤ R}. C 0 = {µ ∈ MF (E) : µ(Km

Then the choice of R and Km imply that / C 0 for some t ≤ T ) < ε. P (XtN ∈ To verify compactness of C 0 , let {µn } be a sequence in C 0 . To find a weakly convergent subsequence we may assume that inf µn (E) > δ > 0. Tightness of {µn /µn (E)} is now clear, and so by Prohorov’s theorem there is a subsequence {nk }over which these normalized measures converge weakly. As the total masses are bounded by R, we may take a further subsequence to ensure convergence of the total masses and hence obtain weak convergence of {µnk }. It follows that K0 = C 0 is compact and so will satisfy (II.4.10). The next step is to show (II.4.12) ∀f ∈ Cb (MF (E)), {f ◦ X N : N ∈ N} is C-relatively compact in D(R).

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Let f ∈ Cb (MF (E)) and ε, T > 0. Choose K 0 as in (II.4.10) and define mi k    , A = h : MF (E) → R : h(µ) = ai µ(fi,j ), ai ∈ R, fi,j ∈ D0 , k, mi ∈ Z+ i=1

j=1

⊂ Cb (MF (E)). Then A is an algebra containing the constant functions and separating points in MF (E). By Stone-Weierstrass there is an h ∈ A so that supµ∈K 0 |h(µ) − f (µ)| < ε. If {Y N } and {Z N } are C-relatively compact in D(R) then so are {aY N + bZ N } and {Y N Z N } for any a, b ∈ R. This is easy to show using (II.4.10) and (II.4.11), for example (but is false for ordinary relative compactness in D(R)). Therefore condition (ii) of the Theorem implies that {h ◦ X N } is C-relatively compact and by (II.4.11) there is a δ > 0 so that lim sup P (w(h ◦ X N , δ, T ) ≥ ε) ≤ ε.

(II.4.13)

N→∞

If s, t ≤ T and |t − s| ≤ δ, then |f (XtN )−f (XsN )| ≤ 2f ∞ 1(X N ([0, T ]) ⊂ K 0 )+2 sup |h(µ)−f (µ)|+|h(XtN )−h(XsN )|, µ∈K 0

and so, w(f ◦ X N , δ, T ) ≤ 2f ∞ 1(X N ([0, T ]) ⊂ K 0 ) + 2ε + w(h ◦ X N , δ, T ). Therefore lim sup P (w(f ◦ X N , δ, T ) ≥ 3ε) N→∞

≤ lim sup P (X N ([0, T ]) ⊂ K 0 ) + lim sup P (w(h ◦ X N , δ, T ) ≥ ε) ≤ 2ε, N→∞

N→∞

the last by (II.4.13) and the choice of K 0 . We have verified (II.4.11) with {f ◦ X N } in place of {X N }, and as (II.4.10) is trivial for this process, (II.4.12) follows. It remains to verify (II.4.11). We may assume d is bounded by 1. Let ε, T > 0, and K 0 is as in (II.4.10). Choose µi ∈ K 0 , i ≤ M , so that K 0 ⊂ ∪M i=1 B(µi , ε), and let fi (µ) = d(µi , µ). Clearly fi ∈ Cb (MF (E)). We showed in the previous paragraph that there is a δ > 0 so that (II.4.14)

M 

lim sup P (w(fi ◦ X N , δ, T ) ≥ ε) ≤ ε.

i=1 N→∞

If µ, ν ∈ K 0 , choose µj so that d(ν, µj ) < ε. Then d(µ, ν) ≤ d(µ, µj ) + d(µj , ν) ≤ |d(µ, µj ) − d(µj , ν)| + 2d(µj , ν) ≤ max |fi (µ) − fi (ν)| + 2ε. i

Superprocesses

159

Let s, t ≤ T , |s − t| ≤ δ. Then the above inequality implies that d(XtN , XsN ) ≤ max |fi ◦ X N (t) − fi ◦ X N (s)| + 2ε + 1(X N ([0, T ]) ⊂ K 0 ), i

and therefore, w(X N , δ, T ) ≤ max w(fi ◦ X N , δ, T ) + 2ε + 1(X N ([0, T ]) ⊂ K 0 ). i

It follows that lim supP (w(X N , δ, T ) ≥ 3ε) N→∞

≤ lim sup P (max w(fi ◦ X N , δ, T ) ≥ ε) + lim sup P (X N ([0, T ]) ⊂ K 0 ) ≤ 2ε, N→∞

i

N→∞

the last by (II.4.14) and the choice of K 0 . This gives (II.4.11) and so the proof is complete. 5. The Martingale Problem In order to prove convergence in Proposition II.4.2 it suffices to show solutions to (M P )X0 are unique in law. We will show this is the case and state the main result of this section in Theorem II.5.1 below. Let g, γ ∈ Cb (E) with γ ≥ 0 as before. A is the weak generator of our BSMP, Y , satisfying (II.2.1) and (II.2.2). Recall that Ag φ = Aφ + gφ. (Ω, F , Ft, P) will denote a filtered probability space with (Ft ) right-continuous. Definition. Let ν be a probability on MF (E). An adapted a.s. continuous MF (E)) iff valued process, X, on (Ω, F , Ft, P) satisfies (LM P )ν (or (LM P )g,γ,A ν t X0 has law ν and ∀φ ∈ D(A) Mt (φ) = Xt (φ) − X0 (φ) −

Xs (Ag φ)ds 0

(LM P )ν t is an (Ft ) − local martingale such that M (φ)t =


Xs (γφ2 )ds. 0

Remark. If X0 (1)dν(X0 ) = ∞, then the integrability of Mt (1) may fail and so we need to work with a local martingale problem. We let (M P )ν denote the corresponding martingale problem (i.e. Mt (φ) is an (Ft )-martingale), thus slightly abusing the notation in Proposition II.4.2. That result shows that if X0 ∈ MF (E), then any limit point of {X N } satisfies (M P )δX0 on the canonical space of measurevalued paths.   Definition. (LM P )ν is well-posed if a solution exists on some Ω, F , Ft, P and the law of any solution (on C(R+ , MF (E)) is unique.     Notation. ΩX = C R+ , MF (E) , FX = Borel sets on ΩX , ΩD = D R+ , MF (E) .   Theorem II.5.1. (a) (LM P )ν is well-posed ∀ν ∈ M1 MF (E) .

160

Superprocesses

(b) There is a family of probabilities {PX0 : X0 ∈ MF (E)} on (ΩX , FX ) such that if Xt (ω) = ωt , then
(i) Pν (·) = PX0 (·)dν(X0 ) is the law of any solution to (LM P )ν for any probability ν on MF (E). (ii) (ΩX , FX , FtX , X, PX0 ) is a BSMP. (iii) If (Zt )t≥0 satisfies (LM P )ν on (Ω, F , Ft, P) and T is an a.s. finite (Ft )  stopping time, then P ZT +· ∈ A | FT (ω) = PZT (ω) (A) a.s. ∀A ∈ FX . (c) If (II.3.1) holds and {X N } are as in Proposition II.4.2, then w

P(X N ∈ ·)⇒PX0 on ΩD .       (d) If Tt F (X0 ) = PX0 F (Xt ) , then Tt : Cb MF (E) → Cb MF (E) . The key step in the above is the uniqueness of solution to (LM P )ν . The remaining properties will be standard consequences of this and the method (duality) used to establish uniqueness (see, e.g., Theorem II.5.6 below). A process satisfying (LM P )ν on (Ω, F , Ft, P ) is called an (Ft )-(A, γ, g)-Dawson-Watanabe superprocess, or (Ft )-(Y, γ, g)-DW superprocess, with initial law ν, or, if ν = δX0 , starting at X0 . The following standard monotone class lemma will be useful. bp

Lemma II.5.2. Let H ⊂ bF be a linear class containing 1 and closed under →. Let H0 ⊂ H be closed under products. Then H contains all bounded σ(H0 )-measurable functions. Proof. See p. 497 of Ethier-Kurtz (1986). Let X satisfy (LM P )ν on some (Ω, F , Ft, P). Let Mloc be the space of continuous (Ft )-local martingales such that M0 = 0. Here processes which agree off an evanescent set are identified. Let P be the σ-field of (Ft )-predictable sets in R+ ×Ω, and define t   Xs (ψs2 γ)ds < ∞ ∀t > 0 , Lloc = ψ : R+ ×Ω×E → R : ψ is P×E−measurable, 2

0

and L = {ψ ∈ Lloc : E 2

2



t

 Xs (ψs2 γ)ds < ∞ ∀t > 0}.

0

A P × E-measurable function ψ is simple (write ψ ∈ S) iff ψ(t, ω, x) =

K−1 

ψi (ω)φi (x)1(ti ,ti+1 ] (t)

i=0

for some φi ∈ D(A), ψi ∈ bFti and 0 = t0 < t1 . . . < tK ≤ ∞. For such a ψ define t Mt (ψ) ≡

ψ(s, x)dM (s, x) = 0

K−1  i=0

  ψi Mt∧ti+1 (φi ) − Mt∧ti (φi ) .

Superprocesses

161

Then a standard argument shows that Mt (ψ) is well-defined (i.e., independent of the choice of representation for ψ) and so ψ → M (ψ) is clearly linear. If ψ˜i (s, ω) = ψi (ω)1(ti ,ti+1 ] (s), K−1
t then ψ˜i is P-measurable and Mt (ψ) = i=0 0 ψ˜i dMs (φi ). Therefore Mt (ψ) is in Mloc and a simple calculation gives t M (ψ)t =

  Xs γψs2 ds.

0

Lemma II.5.3. For any ψ ∈ Lloc there is a sequence {ψn } in S such that  n  P (ψn − ψ)2 (s, ω, x)γ(x)Xs(dx)ds > 2−n  < 2−n . 2

0

Proof. Let S denote the set of bounded P × E-measurable functions which can be bp bp approximated as above. S is clearly closed under →. Since D(A) = bE, S contains ψ(t, ω, x) =

K−1 

ψi (ω, x)1(ti ,ti+1 ] (t)

i=0

where 0 = t0 < . . . < tK ≤ ∞ and ψi (ω, x) = fi (ω)φi (x), φi ∈ bE, fi ∈ bFti . Now apply Lemma II.5.2 to the class H of ψi ∈ b(Fti × E) for which ψ as above is in S. Using H0 = {fi (ω)φi (x) : fi ∈ bFti , φi ∈ bE}, we see that K−1 ψ(t, ω, x) = ψi (ω, x)1(ti ,ti+1 ] (t) is in S for any ψi ∈ b(Fti × E). i=0

If ψ ∈ b(P × E), then −n i2

ψn (s, ω, x) = 2

n

#  ψ(r, ω, x)dr if s ∈ i2−n , (i + 1)2−n

i = 1, 2, . . .

(i−1)2−n

satisfies ψn ∈ S by the above. For each (ω, x), ψn (s, ω, x) → ψ(s, ω, x) for Lebesgue a.a. s by Lebesgue’s differentiation theorem (e.g. Theorem 8.8 of Rudin (1974)) and it follows easily that ψ ∈ S. Finally if ψ ∈ L2loc , the obvious truncation argument and Dominated Convergence (set ψn = (ψ ∧ n) ∨ (−n)) completes the proof. Proposition II.5.4. There is a unique linear extension of M : S → Mloc to a
t map M : L2loc → Mloc such that M (ψ)t = Xs (γψs2 )ds ∀t ≥ 0 a.s. ∀ψ ∈ L2loc . 0

If ψ ∈ L2 , then M (ψ) is a square integrable Ft -martingale.

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Proof. Assume M satisfies the above properties and ψ ∈ L2loc . Choose ψn ∈ S as in Lemma II.5.3. By linearity, n M (ψ) − M (ψn )n = M (ψ − ψn )n =

  2  Xs γ ψ(s) − ψn (s) ds

0

≤ 2−n w.p. > 1 − 2−n . A standard square function inequality and the Borel-Cantelli Lemma imply   sup Mt (ψ) − Mt (ψn ) → 0 a.s. as n → ∞. t≤n

This proves uniqueness and shows how we must define the extension. The details showing that this extension has the required properties are standard and left for the reader. Finally it is easy to use the dominated convergence theorem to see that if M ∈ Mloc satisfies E(M t) < ∞ for all t > 0 then M is a square integrable martingale. This proves the final assertion for ψ ∈ L2 . Remarks. II.5.5. (1) By polarization if φ, ψ ∈ L2loc , t M (φ), M (ψ)t =

Xs (γφs ψs )ds. 0

In particular if A1 and A2 are disjoint sets in E and M (Ai ) = M (1Ai ), then M (A1 ), M (A2 )t = 0 and so Mt is an orthogonal (local) martingale measure in the sense of Ch. 2 of Walsh (1986) where the reader can find more general constructions of this
type. P )ν and use Gronwall’s (2) If X0 (1)dν(X0 ) < ∞, then  φ ≡ 1 in (LM   take and Fatou’s Lemmas to see that E Xt (1) ≤ E X0 (1) eg˜t where g˜ = sup g(x). If ψ x

is P × E measurable and bounded on [0, T ] × Ω × E for all T > 0, then the above shows that ψ ∈ L2 and so Mt (ψ) is an L2 martingale. Let MF denote the Borel σ-field on MF (E). Theorem II.5.6. Assume: (H1 ) ∀X0 ∈ MF (E) there is a solution to (LM P )δX0 .   (H2 ) ∀t ≥ 0 there is a Borel map pt : MF (E) → M1 MF (E) such that if ν ∈ M1 MF (E) and X satisfies (LM P )ν , then P(Xt ∈ A) = pt (X0 , A)dν(X0 ) ∀A ∈ MF . Then (a) and (b) of Theorem II.5.1 hold. Remark. This is a version of the well-known result that uniqueness of the onedimensional distributions for solutions of a martingale problem implies uniqueness in law and the strong Markov property of any solution. The proof is a simple adaptation of Theorem 4.4.2 of Ethier-Kurtz (1986). Note that (H1 ) has already been verified because any limit point in Proposition II.4.2 will satisfy (LM P )δX0 .

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Proof. Let Z satisfy (LM P )ν on (Ω, F , Ft, P) and T be an a.s. finite (Ft )-stopping   time. Choose A ∈ FT so that P(A) > 0 and define Q on (Ω, F ) by Q(B) = P B | A . Let Wt = ZT +t and Gt = FT +t , t ≥ 0. If ν0 = Q(W0 ∈ ·), we claim W satisfies (LM P )ν0 on (Ω, F , Gt, Q). Define 

t+T

Sk = inf t :

 Zr (1)dr > k ∧ k.

T

One easily checks that Sk + T is an (Ft )-stopping time and Sk is a (Gt )-stopping time. Clearly Sk ↑ ∞ P-a.s., and hence Q-a.s. Let M Z be the martingale measure associated with Z, and for φ ∈ D(A) let t MtW (φ)

= Wt (φ) − W0 (φ) −

Ws (Ag φ) ds = MTZ+t (φ) − MTZ (φ). 0

Fix 0 ≤ s < t and D ∈ Gs = Fs+T . The definition of Sk ensures that t Nt =

  1 T ≤ s ≤ Sk + T φ(x)M Z (ds, dx)

0

is an L2 bounded martingale (N t is bounded), and therefore,  W   W Q Mt∧S (φ) − Ms∧S (φ) 1D = P ((NT +t − NT +s ) 1D∩A ) P(A) k k =0 by optional sampling, because D ∩ A ∈ Fs+T . This proves MtW (φ) is a (Gt )-local martingale under Q and a similar argument shows the same is true of t MtW (φ)2 −

Wr (γφ2 )dr. 0

This shows that W satisfies (LM P )ν0 on (Ω, F , Gt, Q). (H2 ) implies that for t ≥ 0 and C ∈ MF , Q(Wt ∈ C) = pt (µ, C)ν0 (dµ), that is

  P (ZT +t ∈ C | A) = E pt (ZT , C) | A ,

and so (II.5.1)

P (ZT +t ∈ C | FT ) = pt (ZT , C) P-a.s.

Therefore {Zt } is (Ft )-strong Markov with Borel transition kernel pt and initial law ν, and hence the uniqueness in II.5.1 (a) is proved.

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(H1 ) and the above allow us to use the above Markov kernel to define the law, PX0 (on ΩX ) of any solution to (LM P )δX0 . (II.5.1) implies X0 → PX0 (A) is Borel for finite-dimensional A and hence for all A in FX . It also implies P (ZT +· ∈ A | FT ) (ω) = PZT (ω) (A) first for A finite-dimensional and hence for all A in FX . Now
consider the “canonical solution” to (LM P )ν , Xt (ω) = ωt , on ΩX under Pν (·) = PX0 (·)dν(X0 ). It is easy to check that X solves (LM P )ν under Pν for any  
t ν ∈ M1 MF (E) . (Note that if Sk = inf{t : Xs (1)ds ≥ k} ∧ k then Mt∧Sk (φ) is a 0   square integrable martingale under PX0 and PX0 Mt∧Sk (φ)2 ≤ γφ2 ∞ k for each X0 ∈ MF (E) and so the same is true the existence part   under Pν .) This proves of Theorem II.5.1(a). By the above ΩX , FX , FtX , X, PX0 is Borel strong Markov, and the proof is complete. To verify (H2 ) we first extend (LM P )ν to time dependent functions. Recall that X satisfies (LM P )ν .  T iff Definition. Let T > 0. A function φ : [0, T ] × E → R is in D(A) (1) For any x in E, t → φ(t, x) is absolutely continuous and there is a jointly Borel ˙ x) = ∂φ (t, x) which is measurable version of its Radon-Nikodym derivative φ(t, ∂t bounded on [0, T ] × E and continuous in x for each t ∈ [0, T ]. (2) For any t ∈ [0, T ], φ(t, ·) ∈ D(A) and Aφt is bounded on [0, T ] × E.  T , then Proposition II.5.7. If φ ∈ D(A) t Xs (φ˙ s + Ag φs )ds ∀t ∈ [0, T ] a.s.

Xt (φt ) = X0 (φ0 ) + Mt (φ) + 0

Proof. Set tni = i2−n and define φn (t, x) = 2n


tni

tn i−1

φ(t, x)dt if tni−1 ≤ t < tni , i ≥ 1.

bp

Clearly φn →φ. It is easy to see φn (t, ·) ∈ D(A) and tni n Aφu (x)du 2n if tni−1 ≤ t < tni . (II.5.2) Aφt (x) = tn i−1

By the (local) martingale problem we have (II.5.3) t t     ds+ Xs Ag φntn Xt φnt = Xtni−1 (φntn )+ φn (s, x)dM (s, x), t ∈ [tni−1 , tni ). i−1

i−1

tn i−1

tn i−1

By the continuity of X we have for i ≥ 2       Xtni−1 φntni−1 = Xtni−1 − φntni−1 − + Xtni−1 φntni−1 − φntni−2 ,

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165

and so for i ≥ 2 and t ∈ [tni−1 , tni ), Xt (φnt )

 = Xtni−1 − φntn



i−1

 + Xtni−1

−

−

φntn i−1

φntn i−2

t



  ds Xs Ag φntn

+

i−1

tn i−1

t φn (s, x)dM (s, x).

+ tn i−1

If t ↑ tni , we get a telescoping sum which we may sum over tni ≤ t (i ≥ 2) and add (II.5.3) with i = 1 and t = tn1 −, together with the above expression for Xt (φnt ) − ∞   1(tni ≤ t)Xtni φntn − φntn , we get Xtni − (φtni − ), where t ∈ [tni , tni+1 ). If Ctn = i

i=1

i−1

t Xt (φnt )

(II.5.4)

=

X0 (φn0 )

+

Ctn

Xs (Aφns + gφns ) ds + Mt (φn ).

+ 0



 Note that if [t] = [2n t]2−n , [t]+ = [2n t] + 1 2−n , then n

ti 

[2n t]

Ctn =

Xtni (φs+2−n − φs ) 2n ds

i=1 tn

i−1 n

ti 

[2n t]

=

i=1 tn

i−1

tn i



[2n t]

=

i=1 tn

i−1

−n s+2

Xtni (φ˙ r )dr ds2n s −n s+2

Xtni (φ˙ r ) − Xr (φ˙ r ) drds2n + s

rXr (φ˙ r ) dr2n 0

[t] +

−n

2

[t] Xr (φ˙ r ) dr +

2−n

+

  Xr (φ˙ r ) [t]+ − r dr2n .

[t]

The sum of the last three terms approach


t

Xr (φ˙ r ) dr for all t ≥ 0 a.s. If

0

hn (r) = sup{|Xu (φ˙ r ) − Xr (φ˙ r )| : |u − r| ≤ 2−n , u ≥ 0}, bp then hn →0 a.s. by the continuity of Xu (φ˙ r ) in u and the first term is at most

[t]

+

hn (r) dr → 0 0

∀t ≥ 0 a.s.

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We have proved that Ctn →


t

Xr (φ˙ r ) dr for all t ≥ 0 a.s. By (II.5.2) we also have

0 n n i ∧t ti [2n t]+1 t

t Xs (Aφns ) ds

=



Xs (Aφu ) du ds2n ,

i=1 tn tn i−1 i−1

0

and an argument very similar to the above shows that t

t Xs (Aφns ) ds

lim

n→∞

=

0

Xs (Aφs )ds

∀t ≥ 0 a.s.

0

By considering M (φ) − M (φn )t we see that P

sup |Mt (φ) − Mt (φn )|→0 as

n → ∞ for all K > 0.

t≤K

The other terms in (II.5.4) converge trivially by Dominated Convergence and so we may let n → ∞ in (II.5.4) to complete the proof. Not surprisingly the above extension is also valid for the martingale problem for Y . ˙ x) + Aφt (x) for (t, x) ∈ [0, T ] × E.  T , let Aφ(t,  Notation. If φ ∈ D(A) x) = φ(t,  T , then Proposition II.5.8. If φ ∈ D(A)

t

Nt = φ(t, Yt ) − φ(0, Y0 ) −

 Aφ(s, Ys ) ds t ∈ [0, T ]

0

is a bounded a.s. cadlag Dt -martingale under P x for all x ∈ E. Its jumps are contained in the jumps of Y a.s. Proof. The continuity properties of φ imply that lim φ(s, Ys ) = φ(t, Yt ) for all t ∈ [0, T ) P x − a.s.

s→t+

and

lim φ(s, Ys ) = φ(t, Yt− ) for all t ∈ (0, T ] P x − a.s.

s→t−

Therefore N is a.s. cadlag on [0, T ] and can only jump at the jump times of Y a.s.  T implies that φ and Aφ  are bounded on [0, T ] × E and hence The definition of D(A) N is also uniformly bounded. Take mean values in Proposition II.5.7 with g ≡ 0 and X0 = δx and use Remark II.5.5 (2) and Exercise II.5.2 (b)(i) below to see that t  s )(x) ds for all (t, x) ∈ [0, T ] × E. Ps (Aφ Pt φt (x) = φ0 (x) + 0

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167

 T −u and so the above implies If u ∈ [0, T ) is fixed then (t, x) → φ(u + t, x) is in D(A) t  s+u ) ds ∀(t, x) ∈ [0, T − u] × E ∀u ∈ [0, T ). Pt φu+t (x) = φu (x) + Ps (Aφ 0

It is now a simple exercise using the Markov property of Y to see that the above implies that Nt is a Dt -martingale under each P x . Green Function Representation  
t  Let Ptg φ(x) = E x φ(Yt ) exp g(Ys ) ds . 0

Exercise II.5.1. (a) Show that Ptg : Cb (E) → Cb (E). Hint: One approach is to use a Taylor series for exp and recall Pt : Cb (E) → Cb (E). bp

(b) Show φ ∈ D(A) ⇔ (Ptg φ−φ)t−1 →ψ ∈ Cb (E) as t ↓ 0, and in this case ψ = Ag φ. d g Pt φ = Ag Ptg φ = Ptg Ag φ ∀φ ∈ D(A). (c) Show that Ptg : D(A) → D(A) and dt The next Exercise will be used extensively. Exercise II.5.2. Assume X solves (LM P )ν . (a) Prove that ∀φ ∈ bE t (GF R)

Xt (φ) =

X0 (Ptg φ)

g φ(x)dM (s, x) Pt−s

+

a.s.

∀t ≥ 0.

0

Hint: Assume first φ ∈ D(A) and apply Proposition II.5.7 for an appropriate choice of φ(s, x).     (b) Assume ν = δX0 . (i) Show that P Xt (φ) = X0 Ptg φ ∀t ≥ 0 and φ ∈ bE. (ii) Show that if 0 ≤ s ≤ t and φ, ψ ∈ bE, s P (Xs (φ)Xt (ψ)) =

X0 (Psg φ) X0

(Ptg ψ)

+

 g  g X0 (Prg γPs−r φPt−r ψ ) dr.

0

Hint: Recall Remark II.5.5. random vector and φ ∈ bE+ , Definition. If W is an  MF (E)-valued  LW (φ) = E(e−W (φ) ) ≡ E eφ (W ) is the Laplace functional of W , or of PW = P (W ∈ ·). bp Lemma II.5.9. Assume D0 ⊂ (bE)+ satisfies D0 = bE+ . Then LW (φ) = LW  (φ) for all φ ∈ D0 implies PW = PW  . Proof. Clearly equality of LW and LW  on D0 implies LW = LW  on bE+ . An elementary argument (see the proof of Proposition 3.4.4 of Ethier-Kurtz (1986)) shows that there is a countable convergence determining set V ⊂ Cb (E)+ (i.e. νn → ν in MF (E) ⇔ νn (φ) → ν(φ) ∀φ ∈ V ). For any φ ∈ V , ν → ν(φ) is measurable with respect to σ(eφ : φ ∈ V ). This implies that the class of open sets in MF (E), and hence MF , is contained in σ(eφ :φ ∈ V ). Apply  the Monotone Class Lemma II.5.2 with H = {Φ ∈ bMF : E Φ(W ) = E Φ(W  ) } and H0 = {eφ : φ ∈ Cb (E)+ } to see that LW = LW  on Cb (E)+ implies PW = PW  . We will verify (H2 ) by giving an explicit formula for LXt . If X and X  are independent solutions of (LM P )δX0 and (LM P )δX  , respectively, then it is easy to 0

168

Superprocesses

check that X + X  satisfies (LM P )δX +X  . This “additive property” is also clear 0 0 for the approximating branching particle systems considered in Proposition II.4.2; the particles do not interact and so superimposing two such systems gives another such system. This leads to LXt +Xt = LXt · LXt and suggests the use of Laplace functionals to characterize the law of Xt . It also explains the “multiplicative” part of the terminology in “critical multiplicative branching measure diffusions”, the catchy name for Dawson-Watanabe superprocesses prior to 1987.  t for a fixed t > 0 and f ∈ Cb (E)+ . By Proposition II.5.7 and Let ψ ∈ D(A) Itˆo’s Lemma for u ≤ t,  exp

u − Xu (ψu ) −

 Xs (f ) ds

0

 = exp − X0 (ψ0 ) − 

u

 exp

s − Xs (ψs ) −

0

u +

 exp

 Xr (f ) dr ψ(s, x)dM (s, x)

0

s − Xs (ψs ) −

0



 Xr (f ) dr −Xs (ψ˙ s + Ag ψs + f − γψs2 /2) ds.

0

Let Nu denote the stochastic integral on the righthand side. Let φ ∈ D(A)+ . Now choose a non-negative ψ so that the drift term vanishes, and ψt = φ, i.e., ψ˙ s + Ag ψs + f − γψs2 /2 = 0, 0 ≤ s ≤ t

(II.5.5)

ψt = φ.

The previous equation then becomes  (II.5.6)

exp

u − Xu (φ) −

   Xs (f ) ds = exp − X0 (ψ0 ) + Nu

u ≤ t.

0

This shows that the local martingale N is bounded and therefore is a martingale satisfying E(Nt ) = 0. Take expectations in (II.5.6) with u = t to see that t    = e−X0 (ψ0 ) dν(X0 ). E exp − Xt (φ) − Xs (f ) ds

(II.5.7)

0

 t iff V ∈ D(A)  t, If a(x, λ) = g(x)λ − γ(x)λ2/2 and Vs ≡ Vsf φ ≡ ψt−s , then ψ ∈ D(A) and (II.5.5) and (II.5.7) (for all t ≥ 0) become, respectively: ∂Vs = AVs + a(·, Vs ) + f, s ≥ 0, ∂s

(SE)φ,f

V0 = φ,

and  (LE)

E exp



t − Xt (φ) −



 Xs (f ) ds

=

f

e−X0 (Vt

φ)

dν(X0 ) ∀t ≥ 0.

0

These arguments trivially localize to t ∈ [0, T ] and hence we have proved:

Superprocesses

169

 T for some Proposition II.5.10. Let φ ∈ D(A)+ and f ∈ Cb (E)+ . If V ∈ D(A) T > 0 and is a non-negative solution V to (SE)φ,f on [0, T ] × E, then (LE) holds for all 0 ≤ t ≤ T for any X satisfying (LM P )ν . The next result will extend this to a larger class of φ and f and solutions to the following mild form of (SE): t (M E)φ,f

Vt = Pt φ +

  Pt−s f + a(·, Vs ) ds.

0

It is easy to check, e.g. by using Proposition II.5.8 to write Vt−s (Ys ) as the sum of a martingale and process of bounded variation, that any solution to (SE) with  t for all t ≥ 0 satisfies (ME). Conversely, a solution to (ME) will V |[0,t]×E ∈ D(A) satisfy (II.5.8)

1 Vt+h − Vt (Ph − I) = Vt + h h h

h

  Pr a(·, Vt+h−r ) + f dr,

0

and hence should satisfy (SE) provided that these limits exist. Theorem II.5.11. Let φ, f ∈ bE+ . (a) There is a unique jointly Borel measurable solution Vtf φ(x) of (M E)φ,f such that V f φ is bounded on [0, T ] × E for all T > 0. Moreover V f φ ≥ 0.  T (b) If, in addition, φ ∈ D(A)+ and f is continuous, then V f φ |[0,T ]×E ∈ D(A) f f ∀T > 0, V˙ t φ(x) and AVt (x) are continuous in t (as well as x), and V f φ satisfies (SE)φ,f . (c) If X is any solution of (LM P )ν then (LE) holds. In view of the locally Lipschitz nature of a(x, λ) in λ, (a) and (b) of the above result are to be expected and will follow from a standard fixed point argument, although some care is needed as a does not have linear growth in λ. The regularity of the solutions in (b) will be the delicate part of the argument. Note that if the spatial motion is Brownian, the argument here may be simplified considerably because of the regularity of the Brownian transition density. (c) follows immediately for φ ∈ D(A)+ and f ∈ Cb (E)+ by Proposition II.5.10. It is then not hard to derive (LE) for all φ, f ∈ bE+ by taking bounded pointwise limits. We defer the details of the proof until the next section and now complete the Proof of Theorem II.5.1. We first verify (H2 ) of Theorem II.5.6. If X satisfies (LM P )δX0 for X0 ∈ MF (E), then (from Theorem II.5.11) for each φ ∈ D(A)+   0 (II.5.9) P e−Xt (φ) = e−X0 (Vt φ) . This uniquely determines the law, pt (X0 , ·), of Xt by Lemma II.5.9. Let X0n → X0 in MF (E), then for any φ ∈ Cb (E), the mean measure calculation in Exercise II.5.2(b) shows that g g n n µ(φ)pt (X0 , dµ) = X0 (Pt φ) → X0 (Pt φ) = µ(φ)pt (X0 , dµ).

170

Superprocesses

This weak convergence shows that if ε > 0 there is a compact subset of E, K,
such that supn µ(K c )pt (X0n , dµ) < ε. This shows that {pt (X0n , ·) : n ∈ N} is tight on MF (E). For example, one can apply Theorem II.4.1 to the set of constant MF (E)-valued processes. (II.5.9) shows that for φ ∈ D(A)+ n −X0n (Vt0 φ) −X0 (Vt0 φ) →e = eφ (µ)pt (X0 , dµ), eφ (µ)pt (X0 , dµ) = e and hence that pt (X0 , ·) is the only possible weak limit point. We have proved that X0 → pt (X0 , ·) is continuous, and in particular is Borel. If X satisfies (LM P )ν , then Theorem II.5.11 (c) shows that   −X0 (Vt0 φ) dν(X0 ) = eφ (µ)pt (X0 , dµ)dν(X0 ) ∀φ ∈ bE+ , P eφ (Xt ) = e and so (H2 ) follows by Lemma II.5.9. This allows us to apply Theorem II.5.6 and infer (a) and (b). The above continuity of pt (X0 , ·) in X0 implies (d). Finally the uniqueness in (a) shows that all the weak limit points in Proposition II.4.2 coincide with PX0 and so the convergence in (c) follows. The following Feynman-Kac formula shows solutions to (M E)φ,f for non-negative φ and f are necessarily non-negative and will be useful in Chapter III. Proposition II.5.12. Suppose φ, f ∈ bE and V : [0, T ] → R is a bounded Borel function satisfying (M E)φ,f for t ≤ T . For u ≤ t ≤ T define u γ(Yr ) Cu = Cu(t) = V (t − r, Y (r)) dr. g(Yr ) − 2 0 Then for all (t, x) ∈ [0, T ] × E, V (t, x) = E x (φ(Yt )eCt ) +

t

E x (f (Yr )eCr ) dr. 0

Proof. Let 0 ≤ s ≤ t. Use (M E)φ,f with t − s in place of t and apply Ps to both sides to derive t−s Ps Vt−s = Pt φ + Pt−r (f + gVr − γVr2 /2) dr 0 s 2 Pr (f + gVt−r − γVt−r /2)dr. = Vt − 0

The Markov Property now shows that s f (Yr ) + g(Yr )Vt−r (Yr ) − γ(Yr )Vt−r (Yr )2 /2 dr, s ≤ t Ns = Vt−s (Ys ) − Vt (Y0 ) + 0

o’s Lemma then implies is a bounded Ds -martingale under P x for all x. Itˆ s s Cs Cr e f (Yr ) dr + eCr dNr . e Vt−s (Ys ) = Vt (x) − 0

0

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171

The stochastic integral is a mean zero L2 -martingale under each P x and so we may set s = t and take means to complete the proof. Extinction Probabilities The Laplace functional equation (LE) is a powerful tool for the analysis of X. As a warm-up we use it to calculate extinction probabilities for X. Assume g(·) ≡ g ∈ R, γ(·) ≡ γ > 0, and X0 ∈ MF (E) is deterministic. Then, setting φ ≡ 1 in (LM P )δX0 , we see that X(1) satisfies the martingale problem characterizing the solution of Xt (1) = X0 (1) +

(II.5.10)

t

t γXs (1)dBs +

0

gXs (1) ds, 0

where B is a linear Brownian motion. An immediate consequence of Theorem w II.5.1(c) is X·n (1)⇒X· (1) in D(R), which when g ≡ 0 reduces to Feller’s Theorem II.1.2. Assume first g = 0 and for λ > 0 let Vt = Vtλ solve ∂Vt γVt2 = AVt − , ∂t 2

V0 ≡ λ.

Clearly the solution is independent of x and so one easily gets Vtλ = 2λ(2 + λtγ)−1 . (LE) implies that the Laplace functional of the total mass of the DW-superprocess is * +   X0 (1)2λ −λXt (1) = exp − (II.5.11) PX0 e . 2 + λtγ Let λ → ∞ to see (II.5.12)

* PX0 (Xt = 0) = PX0 (Xs = 0 ∀s ≥ t) = exp

−2X0 (1) tγ

+ .

In particular, by letting t → ∞ we see that X becomes extinct in finite time PX0 -a.s. See Knight (1981, p. 100) for the transition density of this total mass process. Exercise. II.5.3. Assume γ(·) ≡ γ > 0, g(·) ≡g are constants.   −2X0 (1)g if g = 0. (a) Find PX0 (Xs ≡ 0 ∀s ≥ t). Answer : exp γ(1−e −gt )  1   if g ≤ 0 (b) Show that PX0 (X becomes extinct in finite time) = exp −2X0 (1)g if g > 0 . γ (c) If g > 0 prove that PX0 -a.s. X becomes extinct in finite time or lim Xt (1) = ∞. t→∞

for sufficiently small λ > 0 and Hint. Show that e−λXt (1) is a supermartingale   −λX (1)  −2X0 (1)g t = exp lim EX0 e . γ t→∞

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Exercise II.5.4, Assume X0 ∈ MF (E) − {0}, νN (x, dk) ≡ νN (dk) is independent of x and gN ≡ 0. Prove that for φ bounded and measurable, E(XtN (φ)|XsN (1), s ≥ 0) =

X0 (Pt φ) N X (1). X0 (1) t

Conclude that if X is the (Y, γ, 0)-DW-superprocess (γ ≥ 0 is constant), then E(Xt (φ)|Xs (1), s ≥ 0) =

X0 (Pt φ) Xt (1). X0 (1)

Hint. Condition first on the larger σ-field σ(N α , α ∈ I) ∨ σ(MN ) (recall MN is the Poisson number of initial points). Remark II.5.13. Assume that X satisfies (LM P )ν on (Ω, F , Ft, P), where A is the generator of the d-dimensional symmetric stable process of index α ∈ (0, 2] and the smaller class Cb∞ (Rd ) is used in place of D(A). Recalling that Cb∞ (Rd ) is a core for D(A) (see Example II.2.4), we may pass to the bounded pointwise closure of {(φ, Aφ) : φ ∈ Cb∞ (Rd )} in (LM P )ν by Dominated Convergence. Here note that if bp

Tk = inf{t : Xt (1) > k} ∧ k and φn →φ, then Mt∧Tk (φn ) is a bounded martingale for all n. Therefore X satisfies (LM P )ν and so is an (F )t -(A, γ, g)-DW superprocess with initial law ν. Exercise II.5.5. Let X be the (Y, γ, 0)-DW superprocess starting at X0 ∈ MF (E), where Y is the d-dimensional symmetric stable process of index α ∈ (0, 2] and γ > 0 is constant. For γ0 , λ > 0, let φλ (x) = φ(xλ−1/α ) and (λ)

Xt (φ) =

γ0 Xλt (φλ ), λ

t ≥ 0. (λ)

Prove that X (λ) is a (Y, γγ0, 0)-DW superprocess (starting of course at X0 ). 6. Proof of Theorem II.5.11. Step 1. If φ, f ∈ bE, there is a tmax ∈ (0, ∞] and a unique solution V to (M E)φ,f on [0, tmax ) which is bounded on [0, T ]×E for all T < tmax and satisfies lim Vt  = ∞ t↑tmax

if tmax < ∞. If in addition φ, f ∈ Cb (E) and Pt φ − φ → 0 as t ↓ 0 (as is the case for φ ∈ D(A)), then V : [0, tmax ) → Cb (E) is continuous in norm. This is a standard fixed point argument which only requires a(·, 0) = 0 and   ∀K > 0, a |E×[−K,K] ∈ Cb E × [−K, K] and |a(x, λ) − a(x, λ )| ≤ CK |λ − λ | (II.6.1) for all x in E, λ, λ ∈ [−K, K] for some increasing {CK } and C0 ≥ 1. We start with φ, f as in the second part of the above assertion. We will view f as fixed but will  let φ vary and will choose δ = δ(φ) > 0 below. Define ψ : C [0, δ], Cb (E) → C [0, δ], Cb (E) by t ψ(U )(t) = Pt φ + 0

  Pt−s f + a(Us ) ds.

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173

Note that a(Us ) ≡ a(·, Us (·)) ∈ C  b (E). To see  that ψ does map into the above space, note first that t → Pt φ is in C R+ , Cb (E) by our choice of φ and the semigroup property. If 0 ≤ t < t + h ≤ δ, then   t+h   t        Pt+h−s f + a(Us ) ds − Pt−s f + a(Us ) ds    0 0  h    t         ≤  Pt+h−s f + a(Us ) ds + Pt−s a(Us+h ) − Pt−s a(Us ) ds   0

0

  t ≤ f  + sup a(Us ) h + a(Us+h ) − a(Us )ds s≤δ

→ 0 as

0

h↓0

by (II.6.1) and sup Us  < ∞. Hence ψ is as claimed above. s≤δ   Take K = 2φ + 1 and for U ∈ C [0, δ], Cb (E) define U  = supt≤δ Ut  and let B(0, K) be the set of such U with U  ≤ K. If U ∈ B(0, K) and     0 < δ ≤ ε1 (φ) = K − φ / f  + KCK , then ψ(U ) ≤ φ + δf  + δ sup a(·, λ) ≤ φ + δ[f  + KCK ] ≤ K |λ|≤K

and therefore ψ : B(0, K) → B(0, K). If, in addition, 0 < δ ≤ ε2 (φ) = 1/2CK , then an application of (6.1) shows that for U, V ∈ B(0, K) δ a(Us ) − a(Vs )ds ≤ CK δU − V  ≤

ψ(U ) − ψ(V ) ≤

1 U − V . 2

0

  Now let δ = δ(φ) = min ε1 (φ), ε2 (φ) and note that (II.6.2)

inf

0≤r≤M

δ(r) > 0 for any M > 0.

Then ψ is a contraction on the complete metric space B(0, K) and so has a unique fixed point Vt which solves (M E)φ,f for t ≤ δ. To repeat this construction with Vδ in place of φ we must check that Ph Vδ − Vδ  → 0 as h ↓ 0. Use (M E)φ,f at t = δ to see this reduces to δ 

Ph+δ−s a(Vs ) − Pδ−s a(Vs )ds → 0 0

as h ↓ 0.

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The above norm is at most (0 < h < δ) h

δ Ph+δ−s a(Vs )ds + 

 0

δ−h

  Pδ−s a(Vs+h ) − a(Vs ) ds

Pδ−s a(Vs )ds + 0

δ−h δ−h

≤ 2 sup a(Vs )h + s≤δ

→0

a(Vs+h ) − a(Vs )ds 0

as h ↓ 0

by the norm-continuity of V and (II.6.1). By repeating the previous argument with Vδ in place of φ we can extend V· to a norm-continuous solution to (M E)φ,f on [0, δ1 + δ2 ] where δ1 = δ(φ) and δ2 = δ(Vδ1 ). Continue inductively to ∞ construct a norm-continuous solution to (M E)φ,f on [0, tmax ), where tmax = δn n≥1

and δn+1 = δ(Vδ1 +···+δn ). If tmax < ∞, clearly lim δn = 0 and so (II.6.2) implies lim Vδ1 +···+δn−1  = ∞ and hence lim Vt  = ∞. n→∞ t↑tmax   For φ, f ∈ bE one can use the same existence proof with L∞ [0, δ]×E  , the space  of bounded Borel functions with the supremum norm, in place of Cb [0, δ], Cb (E) . We need the fact Pt φ(x) is jointly Borel which is clear for φ ∈ Cb (E) (because (II.2.2) and (QLC) imply continuity in each variable separately) and hence for all φ ∈ bE by a monotone class argument. It follows easily that ψ(U )(t, x) is Borel and the argument proceeds as above to give a Borel solution of (M E)φ,f on [0, tmax )×E. Turning to uniqueness, assume V and V˜ are solutions to (M E)φ,f on [0, tmax ) and [0, t˜max ), respectively so that V and V˜ are locally (in t) bounded and, in particular, K = sup Vs  ∨ V˜s  < ∞ for t < tmax ∧ t˜max . Then for such a t and s≤t

K,

t Vt − V˜t  ≤ CK

Vs − V˜s ds 0

which implies V = V˜ on [0, tmax ∧ t˜max ) by Gronwall’s Lemma (s → Vs −V s  is universally measurable). Clearly tmax < t˜max is impossible because then lim Vt  = t↑tmax

∞ would imply lim V˜t  = ∞ which is impossible for tmax < t˜max by our local t↑tmax

boundedness assumption on the solution V˜ . Therefore tmax = t˜max and so V = V˜ . This completes Step 1. Step 2. If φ ∈ D(A) and f ∈ Cb (E), then the above solution satisfies the conclusions of (b) for T, t < tmax . The key step will be the existence of ∂V ∂t . In addition to (II.6.1), the only property of a we will use is ∂ a(x, λ) ∈ C(E × R+ ) and satisfies ∂λ lim sup a (λ) − a (λ + δ) = 0 and sup a (·, λ) < ∞ ∀K > 0.

a (x, λ) ≡ (II.6.3)

δ↓0 |λ|≤K

|λ|≤K

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175

Fix 0 < T < tmax . Recall from (II.5.8) that for h > 0, if Rth

−1

h

=h

  Pr f + a(Vt+h−r ) dr,

0

then

Vth

−1

= (Vt+h − Vt )h

satisfies Vth =

(II.6.4)

(Ph − I) Vt + Rth . h

The norm continuity of Vt (from Step 1) and (II.6.1) show that as h, r ↓ 0 (r < h),   bp a(Vt+h−r ) − a(Vt ) → 0 and so Pr f + a(Vt+h−r ) →f + a(Vt ) on [0, T ] × E. It bp

follows that Rth →f + a(Vt ) ∈ Cb ([0, T ] × E) as h → 0+. Therefore it is clear from (II.6.4) that if bp

Vth →V˙ t

(II.6.5)

on

[0, T ] × E

and the limit is continuous in each variable

separately,

then the conclusions of (b) hold on [0, T ] × E and Step 2 is complete. To prove (II.6.5), write Vth

h

Pt+h φ − Pt φ = + h−1 h

  Pt+h−s f + a(Vs ) ds

0

t +

Pt−s

   a(Vs+h ) − a(Vs ) h−1 − a (Vs )Vsh ds +

0

t

  Pt−s a (Vs )Vsh ds

0

(II.6.6) ≡ 1h + 2h + 3h + 4h . The norm continuity of Vs , and hence of a(Vs ), together with φ ∈ D(A) imply (II.6.7)

  bp 1h + 2h →Pt (Aφ) + Pt f + a(φ) on

[0, T ] × E

as h ↓ 0.

Note also that the limit is continuous in each variable if the other is fixed. The mean value theorem shows there is a ζsh (x) between Vs (x) and Vs+h (x) such that " #      a(Vs+h ) − a(Vs ) h−1 − a (Vs )Vsh (x) = a x, ζsh (x) − a (x, Vs (x)) Vsh (x). This together with (II.6.3) and the norm continuity of Vs imply t (II.6.8)

sup |3h | ≤ ηh

Vsh ds for some

x

ηh → 0

as h → 0 + .

0

Our local boundedness condition on a (see (II.6.3)) and norm continuity of V imply t sup |4h | ≤ C

Vsh ds.

x

0

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Use the above bounds in (II.6.6) to get t Vth 

≤C +C

Vsh ds,

t≤T

0

and hence sup Vth  ≤ CeCT .

(II.6.9)

t≤T

We now may conclude from (II.6.8) that sup |3h | → 0 as

(II.6.10)

t≤T,x

h ↓ 0.

The above results and (II.6.6) suggest that V˙ t (if it exists) should solve 



t

Wt = Pt Aφ + f + a(φ) +

(II.6.11)

  Pt−s a (Vs )Ws ds.

0

A slight modification of Step 1 shows there is a unique solution of (II.6.11) in   L∞ [0, T ], Cb (E) . To see this, set θ = Aφ + f + a(φ) ∈ Cb (E) and define ψ : L∞ ([0, T ], Cb (E)) → L∞ ([0, T ], Cb(E)) by

b ψ(W )(t) = Pt θ +

  Pt−s a (Vs )Ws ds.

0

  Clearly h(s, x, λ) = a x, Vs (1) λ is Lipschitz in λ uniformly in (s, x) ∈ [0,  T ]× E and ∞ so as in Step 1 we get the existence of a unique fixed point W first on L [0, δ], Cb (E)  for appropriate δ > 0 and then on L∞ [0, T ], Cb (E) by iteration because the linear growth of h in λ means the solution cannot explode. As Wt (x) is continuous in x for each t and continuous in t for each x (see (II.6.11)), to prove (II.6.5) it suffices to show 

bp

Vth →Wt

(II.6.12)

on

[0, T ] × E.

In view of (II.6.9) we only need establish pointwise convergence. For this we may fix hn ↓ 0 and define r(t, x) = lim |Vthn (x) − Wt (x)| which is bounded on [0, T ] × E n→∞

because W is. Apply (II.6.6), (II.6.7), (II.6.10) and (II.6.11) to see that  t         r(t, x) = lim  Pt−s a (Vs )(Vshn − Ws ) (x)ds n→∞   0

t ≤C

Pt−s (rs )(x)ds, 0

Superprocesses and so

177

t rt  ≤ C

rs ds. 0

This implies r ≡ 0 and hence (II.6.12). The proof of Step 2 is complete. Step 3. If φ, f ∈ bE+ , then tmax = ∞, Vt = Vtf φ ≥ 0 and is bounded on [0, T ] × E ∀T > 0 and (LE) holds if X is any solution of (LM P )ν . The non-negativity is immediate from Proposition II.5.12. For the other assertions assume first φ ∈ D(A)+ , f ∈ Cb (E)+ . Step 2, the non-negativity of V f φ, and Proposition II.5.10 show that (LE) is valid for t < tmax . If g = sup g(x), (M E)φ,f x

and the non-negativity of V f φ imply t Vtf φ

≤ φ + tf  + g

Vsf φds,

t < tmax

0

and therefore Vtf φ ≤ (φ + tf ) egt ,

(II.6.14)

t < tmax .

This means Vtf φ cannot explode at a finite tmax and so tmax = ∞. Turning to more general (φ, f ), let   2 H = (φ, f ) ∈ (bE+ ) : tmax = ∞, (LE) holds . bp

Assume (φn , fn )→(φ, f ) and (φn , fn ) ∈ H. By (II.6.14) we have sup Vtfn φn  < ∞ ∀T > 0.

n,t≤T

bp

Apply (LE) with ν = δδx and (φn , fn ) to see that Vtfn φn →Vt∞ on [0, T ] × E ∀T > 0 (the boundedness is immediate from the above). Now let n → ∞ in (M E)φn ,fn and use Dominated Convergence to see that Vt∞ = Vtf φ and for (φ, f ), tmax = ∞, and (LE) holds by taking limits in this equation for (φn , fn ). This shows H is closed bp

under →. As H ⊃ D(A)+ × Cb (E)+ (by the previous argument) and D(A)+ is bpdense in bE+ (Corollary II.2.3) we may conclude that H = (bE+ )2 . This completes Step 3 because the boundedness claim is immediate from tmax = ∞ and the local boundedness established in Step 1. (a) is immediate from Steps 1 and 3. (c) follows from Step 3. (b) is clear from Step 2 and tmax = ∞ in Step 3.

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7. Canonical Measures Definition. A random finite measure, X, on E is infinitely divisible iff for any natural number n there are i.i.d. random measures {Xi : i ≤ n} such that X and X1 + . . . + Xn have the same law on MF (E). Example. Let (Xt , t ≥ 0) be a (Y, γ, g)-DW-superprocess starting at X0 ∈ MF (E). If {X·i : i ≤ n} are iid copies of the above DW-process but starting at X0 /n, then (II.7.1)

D

X· ≡X·1 + . . . + X·n

as continuous

MF (E)-valued processes.

This follows from Theorem II.5.1, by noting that X 1 + . . . + X n satisfies the martingale problem which characterizes the law of X (or by using the convergence theorem and the corresponding decomposition for the approximating branching particle systems). In particular for each fixed t ≥ 0, Xt is an infinitely divisible random measure. For our purposes, Chapter 3 of Dawson (1992) is a good reference for infinitely divisible random measures on a Polish space (see also Kallenberg (1983) for the locally compact case). The following canonical representation is essentially derived in Theorem 3.4.1 of Dawson (1992). Theorem   II.7.1. Let X be an infinitely divisible random measure on E such that E X(1) < ∞. There is a unique
pair (M, R) such that M ∈ MF (E), R is a measure on MF (E) − {0} satisfying ν(1)R(dν) < ∞, and      (II.7.2) E exp − X(φ) = exp − M (φ) − 1 − e−ν(φ) R(dν) ∀φ ∈ (bE)+ . Conversely if M and R are as above, then the right-hand side of (II.7.2)  is the  Laplace functional of an infinitely divisible random measure X satisfying E X(1) < ∞. Definition. The measure R in (II.7.2) is the canonical measure associated with X. We will give an independent construction of the canonical measure associated with Xt , a DW-superprocess evaluated at t, below (see Theorem II.7.2 and Exercise II.7.1). There are some slight differences between the above and Theorem 3.4.1 of Dawson (1992) and so we point out the necessary changes in the   Proof of Theorem II.7.1. A measure, µ, on E is locally finite µ ∈ MLF (E) iff it is finite on bounded sets. Suppose X is infinitely divisible and E X(1) < ∞. Theorem 3.4.1 of Dawson (1992) shows there is a locally finite measure, M , on E and a measure, R, on MLF (E) − {0} such that (II.7.2) holds for all φ ∈ (bE)+ with bounded support. Fix such a φ. Then for λ > 0   −X(λφ) = λM (φ) + 1 − e−λν(φ) R(dν). (II.7.3) u(λφ) ≡ − log E e
≤ Cλ (1 − e−λν(φ) ) and 1 − e−λν(φ) R(dν) < ∞ for λ > 0, it Since ν(φ)e−λν(φ)
follows that ν(φ)e−λν(φ) R(dν) < ∞ for λ > 0. An application of the Mean Value and Dominated Convergence Theorems allows us to differentiate (II.7.3) with respect to λ > 0 and conclude   −1  −λX(φ) −λX(φ) = M (φ) + ν(φ)e−λν(φ) R(dν). E e E X(φ)e

Superprocesses

179

Let λ → 0+ and use Monotone Convergence to see   (II.7.4) E X(φ) = M (φ) + ν(φ)R(dν) first for φ as above and then for all non-negative measurable φ by Monotone Con
vergence. Take φ = 1 to see M is finite, ν(1)R(dν) < ∞ and so R is supported by MF (E) − {0}. We can also take monotone limits to see that (II.7.2) holds for all φ ∈ (bE)+ . For uniqueness note from (II.7.3) that for any φ in (bE)+   −1 1 − e−λν(φ) λ−1 R(dν) = M (φ), lim u(λφ)λ = M (φ) + lim λ→∞

λ→∞

where in the last line we
used (1 − e−λν(φ) )λ−1 ≤ ν(φ) and Dominated Convergence. This shows that M and h(ν)R(dν) are determined by the law of X for h in C = {h(ν) =

K 

bi e−ν(φi ) : bi ∈ R, φi ∈ (bE)+, h(0) = 0}.

1

Note that for integration purposes h(ν) = −

K

bi (1 − e− ν,φi  ), and C is a vector

1

space closed under multiplication. As in Lemma II.5.9, the Monotone Class Lemma 5.2 shows these integrals determine R. Assume conversely that (II.7.2) holds for some M, R as in the Theorem. As in Theorem 3.4.1 of Dawson the right-hand side is the Laplace functional of some random measure which clearly must  then  be infinitely divisible. One then obtains (II.7.4) as above and this shows E X(1) < ∞. Assume now that X is the (Y, γ, g)-DW superprocess with γ(·) ≡ γ > 0 constant, g ≡ 0, and law PX0 if X starts at X0 ∈ MF (E). Let x0 ∈ E and consider the approximating branching particle systems, X·N , in Theorem II.5.1 starting at δx0 (under PδNx ) and δx0 /N (under PδNx /N ), and with gN ≡ 0 and νN (x, dk) = ν(dk) 0 0 independent of (x, N ). In the former case we start N particles at x0 (see Remark II.3.2) and in the latter we start a single particle at x0 . Let φ ∈ Cb (E)+ and write Vt φ for Vt0 φ, the unique solution of (M E)φ,0. Lemma II.3.3 and Remark II.3.4 (the arguments go through unchanged for our slightly different initial conditions) show that     (II.7.5) N PδNx /N XtN (φ) = PδNx XtN (φ) = Pt φ(x0 ). 0

0

Theorem II.5.1 and (LE) imply that

N      PδNx /N exp −XtN (φ) = PδNx exp −XtN (φ) → exp (−Vt φ(x0 )) as N → ∞. 0

0

Take logarithms and use log z ∼ z −1 as z → 1 (the expression under the N th power must approach 1) to conclude   N 1 − e−Xt (φ) N dPδNx /N = Vt φ(x0 ). (II.7.6) lim N→∞

0

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Superprocesses

Also note by Kolmogorov’s Theorem (II.1.1(a)) that   (II.7.7) lim N PδNx /N XtN = 0 = 2/γt. N→∞

0

(II.7.7) and (II.7.5) easily imply tightness of N PδNx /N (XtN ∈ ·, XtN = 0) and (II.7.6) 0 shows the limit points coincide. The details are provided below. Theorem II.7.2. For each x0 ∈ E and t > 0 there is a finite measure Rt (x0 , ·) on MF (E) − {0} such that w (i) N PδNx /N (XtN ∈ ·, XtN = 0)⇒Rt (x0 , ·) on MF (E) and x0 → Rt (x0 , ·) is Borel 0 measurable, * + (ii) PX0 (exp (−Xt (φ))) = exp −



(iii) Rt (x0 , MF (E) − {0}) = 2/γt,



1 − e−ν(φ) Rt (x0 , dν)dX0 (x0 )

∀φ ∈ bE+ ,

ν(φ)Rt (x0 , dν) = Pt φ(x0 ) ∀φ ∈ bE,

  2 1A (x) exp{−2x/γt}dx ∀A ∈ B (0, ∞) , Rt (x0 , {ν : ν(1) ∈ A}) = (2/γt)
t
(iv) ψ(ν(1))ν(φ)Rt (x0 , dν) = 0 ψ(γtz/2)ze−z dz Pt φ(x0 ) ∀φ ∈ bE, ψ ∈ bB(R+ ). Proof. A sequence {µN } of finite, non-zero measures on MF (E) is tight if supN µN (1) < ∞ and  for any ε > 0 there is a compact set Kε ⊂ E such that µN {ν : ν(Kεc ) > ε} /µN (1) < ε. For example, one may apply Theorem II.4.1 to ˜N ≡ X ˜ N with law µN /µN (1). (II.7.5) and (II.7.7) easily the set of constant paths X 0 imply these conditions for µN (·) = N PδNx /N (XtN ∈ ·, XtN = 0). Let µ∞ be any 0   weak limit point in MF MF (E) . Then (II.7.6) implies (II.7.8) 1 − e−ν(φ) dµ∞ (ν) = Vt φ(x0 ) ∀φ ∈ Cb (E)+ . Take φ ≡ λ > 0 in the above, recall Vt λ = 2λ(2 + λtγ)−1 and let λ → ∞ to see that   µ∞ MF (E) − {0} = 2/γt = lim µN (1) N→∞

(by (II.7.7))

= µ∞ (1). This shows µ∞ ({0}) = 0 and, together with (II.7.8), implies ∀φ ∈ Cb (E)+. (II.7.9) e−ν(φ) dµ∞ (ν) = 2/γt − Vt φ(x0 ) w

As in Lemma II.5.9, this uniquely determines µ∞ and shows µN ⇒µ∞ . The Borel measurability of Rt (x0 ) in x0 is then clear from the Borel measurability of the approximating measures. The proof of (i), (ii) (by (LE)), and the first assertion in (iii) is complete. The second assertion in (iii) is a special case of (iv), proved below. The final assertion in (iii) is obtained by setting φ ≡ λ in (II.7.9), as was already done in the above.

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181

For (iv) it suffices to consider ψ and φ bounded and continuous. For the branching particle system described above, N = σ(N α : α ∈ I) is independent of σ(Y α : α ∈ I) and so N PδNx

0

N N N /N (ψ(Xt (1))Xt (φ)1(Xt (1)



ψ(XtN (1))1(XtN (1) = 0)

= N PδNx

/N

= N PδNx

N N /N (ψ(Xt (1))1(Xt (1)

0

0

= 0))

 1  N Pδx /N (φ(Ytα )|N ) 0 N α∼t

= 0)XtN (1))Pt φ(x0 ).

    Now let N → ∞ in the above. Lemma II.4.6 and N PδNx /N XtN (1)2 ≤ PδNx XtN (1)2 0 0 give us the necessary uniform integrability to use (i) and conclude that ψ(ν(1))ν(φ)Rt (x0 , dν) = ψ(ν(1))ν(1)Rt(x0 , dν)Pt φ(x0 ), and the last part of (iii) completes the proof of (iv). Clearly we have given a direct construction of the canonical measure, Rt (x0 , ·), of Xt under Pδx0 . In this case M ≡ 0. For general γ, g it is not hard to modify the above to recover the canonical representation from our convergence theorem. We leave this as Exercise II.7.1 below. In general M will not be 0 as can readily be seen by taking γ ≡ 0. Exercise II.7.1. Let X be a (Y, γ, g)-DW superprocess starting at δx0 where γ ∈ Cb (E)+ and g ∈ Cb (E). Extend the proof of Theorem II.7.2 to show there is
an Mt (x0 , ·) ∈ MF (E) and a σ-finite measure Rt (x0 , ·) on MF (E) − {0} such that ν(1)Rt (x0 , dν) < ∞ and * + −ν(φ) Rt (x0 , dν) ∀φ ∈ (bE)+. Eδx0 (exp (−Xt (φ))) = exp −Mt (x0 , φ) − 1 − e Hint. Recalling (II.7.6), let ε > 0, φ ∈ Cb (E)+ , and write     N N 1 − e−Xt (φ) N dPδNx /N = N 1 − e−Xt (φ) − XtN (φ) 1(XtN (1) ≤ ε)dPδNx /N 0 0   (∗) + XtN (φ)1 XtN (1) ≤ ε N dPδNx /N 0 N + (1 − e−Xt (φ) )1(XtN (1) > ε)N dPδNx /N . 0

Show that the first term goes to 0 as ε ↓ 0 uniformly in N and that   XtN (·)1(XtN (1) ≤ ε)N dPδNx /N : N ∈ N 0

and

{N PδNx

0

/N

  N Xt ∈ ·, XtN (1) > ε : N ∈ N}

are tight on E and MF (E), respectively. Now let N → ∞ through an appropriate subsequence and then ε = εk ↓ 0 in (∗) to obtain the desired decomposition.

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Superprocesses

Theorem II.7.2 and (II.7.7) imply  w (II.7.10) PδNx /N XtN ∈ · | XtN = 0 ⇒Rt (x0 , ·)/Rt(x0 , 1), 0

that is, Rt (x0 , ·), when normalized, is the law of a cluster at time t of descendants of a common ancestor at t = 0 conditioned on the existence of such descendants. Note that Yaglom’s Theorem II.1.1(b) is immediate from (II.7.10) and Theorem II.7.2(iii). Exercise II.7.2. (a) If X is a (Y, γ, 0)-DW-superprocess under PX0 with γ > 0 w constant, prove that ε−1 Pεδx0 (Xt ∈ ·, Xt = 0)⇒Rt (x0 , ·) as ε ↓ 0 on MF (E). Hint. Use Theorem II.7.2 (ii) to show convergence of the corresponding Laplace functionals.
(b) If Tt is the semigroup of X, show that Rt (x0 , ψ) = Tt−τ ψ(ν)Rτ (x0 , dν) for all 0 < τ ≤ t, x0 ∈ E, and ψ bounded measurable on MF (E) − {0}. Hint. Use (a), first consider ψ ∈ Cb (MF (E)) such that ψ(0) = 0, and recall that Tt : Cb (MF (E)) → Cb (MF (E)). If X0
∈ MF (E), let Ξt,X0 be a Poisson point process on MF (E) − {0} with intensity Rt (x0 , ·)dX0 (x0 ). Theorem II.7.2(ii) implies that (II.7.11) ν Ξt,X0 (dν) is equal in law to PX0 (Xt ∈ ·). In view of (II.7.7) and (II.7.10) we see that this representation decomposes Xt according to the Poisson number of ancestors at time 0 with descendants alive at time t. This perspective will allow us to extend Theorem II.7.2 and this Poisson decomposition to the sample paths of X. Indeed, (II.7.1) shows that infinite divisibility is valid on the level of sample paths. Let ζ : ΩD → [0, ∞] be given by ζ(X) = inf{t > 0 : Xt = 0} and define ΩEx = {X ∈ ΩD : X0 = 0, ζ > 0, Xt ≡ 0 ∀t ≥ ζ} , . / Ex ΩEx : X· is continuous ⊂ ΩX , C = X ∈Ω equipped with the subspace topologies they inherit from ΩD and ΩX , respectively. If w {Nk : k ∈ N∪{∞}} are measures on ΩEx we write Nk ⇒N∞ on ΩEx if Nk (ζ > t) < ∞ for all k ∈ N ∪ {∞} and t > 0, and w

Nk (X ∈ ·, ζ > t)⇒N∞ (X ∈ ·, ζ > t) as k → ∞, as finite measures on ΩD ∀t > 0. Theorem II.7.3. (a) For each x0 ∈ E there is a σ-finite measure, Nx0 , on ΩEx C w such that N Pδx0 /N (X·N ∈ ·)⇒Nx0 on ΩEx . (b) For all t > 0, Nx0 (Xt ∈ ·, ζ > t) = Rt (x0 , ·). with intensity Nx0 . Then (c) Let
Ξ be a Poisson point process on ΩEx C Xt = νt dΞ(ν), t > 0, has the law of a (Y, γ, 0)-DW-superprocess starting at δx0 . Remark II.7.4. Note that (II.7.7) and the equality Nx0 (ζ > t) = Rt (x0 , 1) = 2/γt allow us to use (a) to conclude w  Pδx0 /N X N ∈ · | XtN = 0 ⇒Nx0 (X ∈ · | ζ > t) on ΩD ∀t > 0.

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183

In this way Nx0 describes the time evolution of a cluster starting from a single ancestor at x0 given that it survives for some positive length of time. We call Nx0 the canonical measure of the process X. It has been studied by El Karoui and Roelly (1991) and Li and Shiga (1995). A particularly elegant construction of Nx0 in terms of Le Gall’s snake may be found in Chapter IV of Le Gall (1999). The reader may want to skip the proof of Theorem II.7.3 on a first reading. δYtα Proof. Let X·N be as before under PδNx , and for i ≤ N let XtN,i = N1 0

α∼t,α0 =i th N be the portion of Xt descending from the i initial ancestor. Fix t > 0 and set N,i ΛN = 0}. The mutual independence of Gi = σ(Y α , N α : α > i), t = {i ≤ N : Xt N,i : i ∈ ΛN i = 1 . . . , N , shows that conditional on ΛN t , {X· t } are iid with law N N N Pδx /N (X· ∈ · | Xt = 0). We also have 0 N Xt+· =

(II.7.12)



N,i Xt+· .

i∈ΛN t N N = 0)) and so by (II.7.7), Clearly |ΛN t | = card(Λt ) is binomial (N, Pδx0 /N (Xt converges weakly to a Poisson random variable Λt with mean 2/γt. The left side of (II.7.12) converges weakly on ΩD to Pδx0 (Xt+· ∈ ·) (use the fact that the limit is a.s. continuous) and so for ε > 0 there is a compact set Kε ⊂ ΩD such that N ∈ Kεc ) < ε for all N . Use (II.7.12) to see that this means that for all N PδNx (Xt+· 0

ε > PδNx

0



 N N Xt+· ∈ Kεc , |ΛN t | = 1 = Pδx



/N 0

 N Xt+· ∈ Kεc |XtN = 0 P (|ΛN t | = 1).

N ∈ · | XtN = 0) : N ∈ N} on ΩD because This proves tightness of {PδNx /N (Xt+· 0  N  lim P |Λt | = 1 = (2/γt)e−2/γt > 0. Let P t be any limit point on ΩD . If Λt is N→∞

the above Poisson random variable and conditional on Λt , {X·i : i ≤ Λt } are iid with law P t , then we may let N → ∞ through an appropriate subsequence in (II.7.12) and conclude Λt 

(II.7.13)

X·i

has law

Pδx0 (Xt+· ∈ ·).

i=1

Note that P (Λt ≥ 1) = 1 − e−2/γt = Pδx0 (Xt = 0) (recall (II.5.12)). From this and Λt X·i = 0 iff Λt = 0 and therefore the above we may conclude that i=1

 P

Λt ≥ 1,

Λt 

 X·i

∈·

= Pδx0 (Xt+· ∈ ·, Xt = 0) .

i=1

The measure on the right is supported on ΩX = {X ∈ ΩX : X0 = 0, ζ > 0, Xs = 0 all s ≥ ζ}, and so the same must be true of P t = L(X·i ) as P (Λt = 1, X01 ∈ ΩX ) = 0 by the above.

184

Superprocesses If 0 ≤ t1 < . . . < tk and φi ∈ Cb (E)+ for 1 ≤ i ≤ k, (II.7.13) shows that

k k          Pδx0 exp − Xtj +t (φj ) = exp − 1 − exp − νtj (φj ) dP t (ν)2/γt . 1

1

This uniquely determines




k   exp − νtj (φj ) dP t (ν) and hence the finite-dimensional 1

distributions of P t by a now familiar monotone class argument. We have shown (use (II.7.7))   N ∈ ·, XtN = 0 N PδNx /N Xt+· 0   N   ∈ · | XtN = 0 = N PδNx /N XtN = 0 PδNx /N Xt+· (II.7.14) 0 0 w 2 ⇒ Pt on ΩD , γt where the limit is supported on ΩX . To handle the small values of t we need a definition and a Lemma. Notation. Lip1 = {φ : E → R : φ ≤ 1, |φ(x) − φ(y)| ≤ ρ(x, y)} (ρ is a fixed complete metric on E). Definition. The Vasershtein metric d = dρ on MF (E) is d(µ, ν) = sup {|µ(φ) − ν(φ)| : φ ∈ Lip1 } . Then d is a complete metric which induces the weak topology on MF (E) (e.g. see Ethier and Kurtz (1986), p. 150, Exercise 2). It only plays an incidental role here but will be important in treating models with spatial interactions in Chapter V. Redefine X·N near t = 0 by * N Xt if t ≥ N −3 N ˜ . Xt = 3 N tN XN −3 if t ∈ [0, N −3 )   Lemma II.7.5. (a) N PδNx /N sup XsN (1) > ε ≤ 4γδε−2 for all δ, ε > 0 and 0

s≤δ

N ≥ 2/ε. (b) There are N0 ∈ N and δ0 > 0 such that  ˜ N (1) > δ 1/5 ≤ δ 1/2 for N PδNx /N sup X s 0

(c) N PδNx

0

s≤δ

   N ˜N ≥ 4/N /N sup d Xt , Xt t

0 < δ ≤ δ0

and N ≥ N0 .

≤ γN −1 .

Proof. (a) Use (M P )N in Section II.4 and (II.4.5) with φ = 1, gN = 0 and γN constant to see that under PδNx /N , XtN (1) is a martingale with predictable square 0
t N function γ Xs (1)ds. The weak L1 inequality and Lemma II.3.3 (a) imply that for 0

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185

N > 2/ε N PδNx

0

 N /N sup Xs (1) > ε s≤δ

 ≤ N PδNx

0

sup XsN (1) − X0N (1) ≥ ε/2

/N

s≤δ



≤ N 4ε−2 PδNx

0

/N

γ

δ

 XsN (1)ds = 4γδε−2 .

0

(b) Assume first δ ≥ N

−3

. Then for N ≥ N0 , 2δ

−1/5

 N PδNx

0

/N

˜ N (1) > δ 1/5 sup X s s≤δ

= N PδNx

0



/N

≤ N and so by (a) for δ < δ0 ,  sup

N −3 ≤s≤δ

XsN (1) > δ 1/5

≤ 4γδ 3/5 ≤ δ 1/2 . Assume now δ < N −3 . Then the above probability equals     N 1/5 N −4/5 −3 N N PδNx /N δN 3 XN = N PδNx /N XN . −3 (1) > δ −3 (1) > δ 0

0

Our assumption on δ implies 2δ 4/5 N 3 < 2N 3/5 < N for N ≥ N0 and so (a) bounds the above by 4γN −3 N 6 δ 8/5 < 4γδ 3/5 < δ 1/2 for δ ≤ δ0 . (c) If f ∈ Lip1 and t < N −3 ,   # "  N ˜ tN (f ) ≤ f  XtN (1) + tN 3 X N−3 (1) ≤ 2 sup XtN (1). Xt (f ) − X N t≤N −3

˜ N ) and an This shows that the right-hand side is an upper bound for supt d(XtN , X t application of (a) completes the proof. We now complete the Proof of Theorem II.7.3. Lemma II.7.5 (c) and (II.7.14) show that if tn ↓ 0 ˜ nε compact in ΩD such that (tn > 0) is fixed and ε > 0, we may choose K   ˜ N ∈ K ˜ ε , X N = 0 < ε2−n . (II.7.15) sup N PδNx /N X tn +· n tn N

0

For t > 0 define . ˜ ε ∀tn ≤ t and Ktε = X ∈ ΩD : Xtn +· ∈ K n

/ sup Xs (1) ≤ 2−2n/5 for all n ≥ 1/ε .

s≤2−2n

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Superprocesses

Lemma II.7.5(b) and (II.7.15) show that for ε < ε0 and N ≥ N0   ˜ N ∈ K ε , X N = 0 N PδNx /N X t t 0    ˜ N ∈ K ˜ ε , X N = 0 1(tn ≤ t)N PδNx /N X ≤ tn +· n tn 0

n

+



N PδNx

n≥1/ε

≤

∞ 

ε2−n +

n=1



/N 0



˜ N (1) > 2−2n/5 sup X s



s≤2−2n

2−n ≤ 2ε.

n≥1/ε

It is a routine Skorokhod space exercise to check that Ktε is a compact subset of Ω0D = {X ∈ ΩD : X0 = 0}. This together with Lemma II.7.5(c) shows that {N PδNx /N (X N ∈ ·, ζ > t) : N ∈ N} is relatively compact in ΩD and all limit points 0 are supported on Ω0D . Fix t0 > 0. Choose Nk → ∞ such that  w (II.7.16) N PδNx /N X N ∈ ·, ζ > t0 ⇒Nt0 on ΩD as N → ∞ through {Nk }. 0

To ease our notation we write N for Nk and QN for N PδNx /N . By taking a further 0 subsequence we may assume (II.7.16) holds with tm in place of t0 (recall tm ↓ 0). Clearly Ntm (·) are increasing in m and so we may define a measure on ΩD by Nx0 (A) = lim Ntm (A). Let tm < t. Theorem II.7.2 implies that m→∞

  lim lim QN XtN (1) ∈ (0, ε), ζ > tm ≤ lim Rt (x0 , {ν : ν(1) ∈ (0, ε)}) = 0. ε↓0 N→∞

ε↓0

A standard weak convergence argument now gives     lim QN X N ∈ ·, ζ > t = lim QN X N ∈ ·, XtN (1) > 0, ζ > tm N→∞

N→∞

= Ntm (· , Xt (1) > 0) . This shows that the measure on the right side is independent of m and so the above implies w  ∀t > 0, (II.7.17) QN X N ∈ ·, ζ > t ⇒Nx0 (· , Xt (1) > 0) and in particular (take t = tm ) Nx0 (·, Xtm (1) > 0) = Ntm (· ), m = 0, 1, 2, . . . .   (II.7.17) shows that the measures Nx0 · , Xt (1) > 0 are decreasing in t and this implies for each s < t, Nx0 a.s. Xs = 0 implies Xt = 0. Right-continuity implies

(II.7.18)

Xs = 0 ⇒ Xt = 0 ∀s < t Nx0 -a.s. ! This implies {ζ = 0} ⊂ {Xtm = 0} Nx0 -a.s., and therefore Ntm -a.s. Therefore (II.7.19)

m

Ntm (ζ = 0) ≤ Ntm (Xtm = 0) = 0,

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187

the last by (II.7.18). It follows that Nx0 (ζ = 0) = lim Ntm (ζ = 0) = 0 which, m→0

together with (II.7.19) shows that Nx0 is supported by ΩEx . (II.7.17) may therefore be written as  w ∀t > 0. (II.7.20) QN X N ∈ ·, ζ > t ⇒Nx0 (·, ζ > t) The convergence in (II.7.14) and the above together imply (II.7.21)

Nx0 (Xt+· ∈ ·, ζ > t) =

2 t P (·) γt

(Note that (II.7.20) alone would not give this if t is a point of discontinuity, but as the limit in (II.7.14) exists we only need identify the finite-dimensional distributions in terms of Nx0 and (II.7.20) is enough for this.) This implies Nx0 -a.s. continuity of Ex Xt+· for all t > 0 (recall P t (Ωc X ) = 0) and hence shows Nx0 is supported on ΩC . (II.7.21) also identifies the finite dimensional distributions of Nx0 and so by (II.7.18) with m = 0 we may conclude that all limit points in (II.7.16) equal Nx0 (·, ζ > t0 ). This proves (a). (b) is then immediate from   Theorem II.7.2(i). Let Ξ be as in (c). Note that Ξ {ν : νt = 0} is Poisson with mean Λt
D i Xt+· , where Λt is Poisson Nx0 (ζ > t) = 2/γt (by (II.7.21)) and so νt+· Ξ(dν)≡ i=1

i (2/γt) and given Λt , {Xt+· : i ≤ Λt } are iid with law Nx0 (Xt+· | ζ > t) = P t (by (II.7.21)). Compare this with (II.7.13) and let t ↓ 0 to complete the proof of (c).

8. Historical Processes. We return to the path-valued setting of Example II.2.4(c) under the assumption (P C)

x → Px

is continuous.

ˆ ˆ In addition tothe E-valued BSMP  Wt with laws (Pτ,y ) described there, we introduce ˆ on D(E) by probabilities Pτ,y : (τ, y) ∈ E Pτ,y (A) = P y(τ ) ({w : (y/τ /w) ∈ A}) . If W· has law Pˆτ,y and Y has law Pτ,y then (II.8.1)

 D (Wt )t≥0 ≡ τ + t, Y τ +t t≥0 .

ˆ γˆ ∈ Cb (E) ˆ + , and for τ ≥ 0 define Let gˆ ∈ Cb (E),   . / MFτ (D) = m ∈ MF D(E) : y τ = y m − a.a. y . ˆ and let X ˆ be the (W, γˆ , gˆ)-DW superproFix m ∈ MFτ (D) so that δτ × m ∈ MF (E) ˆ ˆ cess starting at δτ ×m with law Pτ,m (≡ Pδτ ×m ) on the canonical space of continuous ˆ MF (E)-valued paths. Introduce      ΩH [τ, ∞) = H· ∈ C [τ, ∞), MF D(E) : Ht ∈ MFt (D) ∀t ≥ τ ,

188

Superprocesses

ˆ → D(E) be the and let ΩH = ΩH [0, ∞) with itsBorel σ-field FH . Let Π : E projection map and define an MF D(E) -valued process (Ht , t ≥ τ ) by   ˆ t Π−1 (A) . Hτ +t (A) = X ˆ τ,m -a.s. ˆ t = δτ +t × Hτ +t ∀t ≥ 0 and Ht ∈ M t (D) ∀t ≥ τ P Lemma II.8.1. X F t 
 Proof. Let Pˆtgˆ f (τ, y) = Pˆτ,y exp{ gˆ(Ws ) ds}f (Wt) and 0

ˆ : u = τ + t}. Λ(t) = {(u, y) ∈ E Then by Exercise II.5.2(b)    ˆ t Λ(t) = Pˆ gˆ (1Λ(t) )(τ, y) dm(y) = 0 Pτ,m X t   ˆ t = δτ +t × Hτ +t Pτ,m -a.s. for because Wt = τ + t, Y τ +t Pˆτ,y -a.s. This shows X each t ≥ 0 and hence for all t ≥ 0 a.s. by the right-continuity of both sides. Since ˆ t ∈ MF (E) ˆ ∀t ≥ 0 a.s., the second assertion follows immediately. X The process of interest is the MF (D)-valued process H. We abuse our notation and also use Ht to denote the coordinate variables on ΩH and let F H [s, t+] =

∞ 

σ(Hr : s ≤ r ≤ t + 1/n).

n=1

  ˆ τ,m (H ∈ ·), where H is as in Lemma Define Qτ,m on ΩH , F H [τ, ∞) by Qτ,m (·) = P ˆ II.8.1. The fact that X is a BSMP easily shows that H ≡ (ΩH , FH , F H [τ, t+], Ht, Qτ,m ) is an inhomogeneous Borel strong Markov process (IBSMP) with continuous paths in MFt (D) ⊂ MF (D). This means (i) ∀u > 0 and A ∈ F H [u, ∞) (τ, m) → Qτ,m (A) is Borel measurable on {(τ, m) : m ∈ MFτ (D), τ ≤ u}. Qτ,m -a.s. (ii) Hτ = m, Ht ∈ MFt (D) ∀t≥ τ , and H· is continuous  (iii) If m ∈ MFτ (D), ψ ∈ bB [τ, ∞] × MF (D) and T ≥ τ is an (F H [τ, t+])t≥τ stopping time, then      Qτ,m ψ(T, HT +· ) | F H [τ, T +] (ω) = QT (ω),HT (ω) ψ T (ω), HT (ω)+· Qτ,m − a.s. on

{T < ∞}.

ˆ is a BSMP (only (iii) requires This is a simple consequence of the fact that X a bit of work) and the routine proof is contained in Dawson-Perkins (1991) (Proof of Theorem 2.1.5 in Appendix 1). We call H the (Y, γˆ , gˆ)-historical process. Of course it is now a simple matter to interpret the weak convergence theorem (Theorem II.5.1(c)), local martingale problem (LM P )ν , and Laplace equation (LE) ˆ in terms of H. for X, To link the weak convergence result with for the (Y,  that   γ, g)-superprocess consider the special case where γˆ (t, y) = γ y(t) , gˆ(t, y) = g y(t) for some g ∈

Superprocesses

189

Cb (E) and γ ∈ Cb (E)+ , τ = 0 and m ∈ MF0 (D). Note that we can, and shall, ˆ consider m as a finite measure on E. Note also that γˆ and gˆ are continuous on E (but not necessarily on R+ × D) – see Exercise II.8.1below. It is then natural to   assume our approximating branching mechanisms νˆN (t, y), · = ν N y(t), · where {ν N } satisfy (II.3.1). Let {Y α : α ∼ t} be the system of branching Y -processes α constructed in Section II.3. If Wtα = (t, Y·∧t ), then {W α : α ∼ t} is the analogous system of branching W -processes and so Theorem II.5.1(c) implies (II.8.2)

 w ˆ ˆ N (·) = 1 δW α ⇒X X t t N α∼t t

  ˆ , on D R+ , MF (E)

ˆ has law P0,m . where X     ˆ N Π−1 (·) , i.e., Recall HtN ∈ MF D(E) is defined by HtN = X t HtN =

1  δY α , N α∼t ·∧t

and so, taking projections in (II.8.2), we have (II.8.3)

w

P(H·N ∈ ·)⇒Q0,m (·).

Remark II.8.2. It is possible to prove this weak convergence result without the continuity assumption (PC) and to prove Theorem II.5.1(c) with the assumption (II.2.2) Pt : Cb → Cb replaced by the weaker condition (QLC) (i.e. Y is a Hunt process). For γ constant and g ≡ 0 these results are proved in Theorems 7.15 and 7.13, respectively, of Dawson-Perkins (1991). Our proof of the compact containment condition (i) can be used to simplify the argument given there. Without our continuity assumptions one must work with the fine topology and use a version of Lusin’s theorem. The processes of interest to us satisfy our continuity conditions and so we will not discuss these extensions. It is a relatively simple matter to take projections in (II.8.2) (or (II.8.3)) and compare with Theorem II.5.1(c) to see that (II.8.4) Xt ≡ Ht (yt ∈ ·) is a (Y, γ, g)-DW superprocess starting at m under Q0,m . We leave this as Exercise II.8.1. See Exercise II.8.3 for another approach. ˆ : R+ × D(E) → E by Π(t, ˆ y) = y(t). Show that Π ˆ is Exercise II.8.1. (a) Define Π ˆ is. not continuous but its restriction to E ˆ Π(t, ˆ y) = y(T ) for all T ≥ t. Hint. On E, ˜ ˜ : ΩH → Ω X = Ht (yt ∈ ·) ∈ MF (E). Show that Π (b) For H ∈ ΩH define Π(H)(t) and is continuous. Hints. (i) Show ˆ T :ΩH → C(R+ , MF (E)) Ht → δt × Ht is continuous. ˆ −1 . ˜ (ii) Show that Π(H) t = T (H)t ◦ Π

190

Superprocesses

˜ ·) (c) Use either (II.8.2) and (a), or (II.8.3) and (b), to prove that under Q0,m , Π(H is a (Y, γ, g)-DW superprocess starting at m ∈ MF (E). Consider now the version of (MP) which characterizes Qτ,m . For s ≤ t and φ      in bD D = B D(E) let Ps,t φ(y) = Ps,y φ(Y t ) be the inhomogeneous semigroup associated with the path-valued process Y t . If Aˆ is the weak generator of W and ˆ ˆ Aφ(s, y) ≡ Aˆs φ(y), then it is easy to see from Proposition II.2.1 that for φ ∈ Cb (E), bp ˆ ˆ ⇔ (Ps,s+h φs+h − φs ) /h→ Aˆs φ as h ↓ 0 for some Aˆs φ(y) in Cb (E) φ ∈ D(A) t ˆ φ(t, Y t ) − φ(s, Y s ) − Aˆr φ(Y r ) dr, ⇔ For some Aˆs φ(y) in Cb (E), s

ˆ t ≥ s, is a Ps,y -martingale ∀(s, y) ∈ E. If m ∈ MFτ (D), an (Ft )t≥τ -adapted process (Ht )t≥τ with sample paths in ΩH [τ, ∞) defined on (Ω, F , (Ft)t≥τ , P) satisfies (HM P )τ,m iff Hτ = m a.s. and t ˆ ∀φ ∈ D(A)

Hs (Aˆs φ + gˆs φs ) ds is a

Mt (φ) = Ht (φt ) − Hτ (φτ ) − τ

t continuous (Ft )-martingale with M (φ)t =

Hs (ˆ γs φ2s ) ds ∀t ≥ τ

a.s.

τ

ˆ As we are considering The following is immediate from Theorem II.5.1, applied to X. deterministic initial conditions we may work with a martingale problem rather than a local martingale problem (recall Remark II.5.5(2)). Theorem II.8.3. (a) (HM P )τ,m is well-posed. Qτ,m is the law of any solution to (HM P )τ,m . (b) If K satisfies (HM P )τ,m on (Ω, F , (Ft)t≥τ P) and T ≥ τ is an (Ft )t≥τ -stopping time, then P(KT +· ∈ A | FT )(ω) = QT (ω),KT (ω) (HT (ω)+· ∈ A) a.s. ∀A ∈ FH . We call a solution (Kt , t ≥ τ ) on (Ω, F , (Ft)t≥τ , P), an (Ft )-historical process (or (Ft ) − (Y, γˆ , gˆ)-historical process) starting at (τ, m). The Feyman-Kac semigroup associated with Y t and gˆ is g ˆ φ(y) Ps,t



 t

= Ps,y exp

  gˆ(u, Y u ) du φ(Y t )

0 ≤ s ≤ t,

φ ∈ bD.

s

The mean measure formula for DW-superprocesses (Exercise II.5.2 (b)) gives    t    ˆ ˆ ˆ gˆ(Ws ) ds φ(Wt ) dm(y), Pτ,m Xt (φ) = Pτ,y exp 0

Superprocesses

191

which by (II.8.1) and Lemma II.8.1 gives (set φ(t, y) = ψ(y t) for ψ ∈ bD) g ˆ ψ(y) dm(y) t ≥ τ, ψ ∈ bD. (II.8.5) Qτ,m (Ht (ψ)) = Pτ,t ˆ fˆ ∈ bEˆ+ (Eˆ is the Borel σ-field Let {Pˆt } denote the semigroup of W and let φ, ˆ ˆ of E). The Laplace equation for X (Theorem II.5.11) shows that if Vˆt is the unique solution of  2 t γˆ (Vˆt−s ) ˆ ˆ ˆ ˆ ˆ ˆ 0 Vt = Pt φ + Ps f + gˆVt−s − (M E)φ, ds, ˆ fˆ 2 0

then 0 (LE)

* + t    ˆ ˆ ˆ ˆ ˆ ˆ Pτ,m exp −Xt (φ) − Xs (f ) ds = exp − Vt (τ, y) dm(y) . 0

Let Ds = {y ∈ D(E) : y = y s }. Defining Vs,t (y) = Vˆt−s (s, y) (s ≤ t, y ∈ Ds ) 0 establishes a one-to-one correspondence between solutions of (M E)φ, ˆ fˆ and solutions of t (M E)φ, ˆ fˆ

Vτ,t (y) = Pτ,t (φˆt )(y) +

  2 Pτ,s fˆs + gˆs Vs,t − γˆs Vs,t /2 (y) ds.

τ τ ˆ Note that in (M E)φ, ˆ fˆ we may fix t (and φt ) and obtain an equation in y ∈ D , τ ≤ t. ˆ y) = φ(y t ) for φ ∈ bD+ , and fˆ(t, y) = f (t, y t) Using Lemma II.8.1, and setting φ(t, 0 into for f ∈ b(B(R+ ) × D) , we readily translate (LE) +

Theorem II.8.4. Assume φ(y) and f (t, y) are non-negative, bounded, Borel functions on D(E) and R+ × D(E), respectively. Let Vτ,t (y) be the unique solution of ˆ ˆ (M E)φ, ˆ fˆ with φ and f as above. Then 

(HLE)

Qτ,m exp



t − Ht (φ) −

 Hs (fs ) ds

* + = exp − Vτ,t (y) dm(y) .

τ

Exercise II.8.2. Assume (Ht , t ≥ τ )
is an (Ft )−(Y, γˆ , 0)-historical process starting at (τ, m). Show that for any φ ∈ bD, φ(y τ )Ht (dy)is a continuous (Ft )-martingale. Hint. The martingale property is easy (use (II.8.5)). For the continuity, start with φ continuous and then show the class of φ for which continuity holds is closed under bp →. Exercise II.8.3. Let m ∈ MFτ (D), and assume gˆ(s, y) = g(y(s)) and γˆ (s, y) = γ(y(s)) for some g ∈ Cb (E) and γ ∈ Cb (E)+. Let K satisfy (HM P )τ,m on (Ω, F , (Ft)t≥τ , P ). Define Xt ∈ MF (E) by Xt (A) = Kτ +t ({y : yτ +t ∈ A}). From the hint in Exercise II.8.1(a) it is easy to see that X· is a.s. continuous.

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Superprocesses

ˆ y) = φ(y(s)), (s, y) ∈ E, ˆ defines a function in D(A) ˆ (a) If φ ∈ D(A), prove that φ(s, ˆ and Aφ(s, y) = Aφ(y(s)). and conclude that P (X ∈ ·) = PX0 is the law of (b) Show that X solves (M P )g,γ,A X0 the (Y, γ, g)-superprocess starting at X0 = m(yτ ∈ ·). ˆ t is an infinitely divisible random measure, the same is true of Ht Since X under Qτ,m for each t ≥ τ . We can therefore introduce the canonical measures from Section 7 in the historical setting. Assume γˆ ≡ γ is constant and gˆ ≡ 0. If ˆ t from Theorem II.7.2. ˆ let R ˆ t (τ, y) denote the canonical measure of X (τ, y) ∈ E, ˆ Then Lemma II.8.1 and (II.7.11), applied to X, imply that ˆ t (τ, y)(·) = δτ +t × Rτ,τ +t(y, ·), R where (by Theorem II.7.2) (a) Rτ,t (y, ·) is a finite measure on MFt (D) − {0} which is Borel in 2 , y ∈ Dτ , and satisfies Rτ,t (MFt (D) − {0}) = γ(t − τ ) ∞ ψ(γ(t − τ )z/2)ze−z dzPτ,y (φ(Y t )) ψ(ν(1))ν(φ)Rτ,t(y, dν) = 0

for any bounded Borel φ : D(E) → R and ψ : R+ → R, and Rτ,t (y, {w : wτ = y}) = 0.  1 − e−ν(φ) Rτ,t(y, dν)m(dy) (II.8.6) (b) Qτ,m (exp(−Ht (φ))) = exp − for any Borel φ : D(E) → R+ . (c) If m ∈ MFτ (D) and Ξ is a Poisson point process on MF (D) − {0} νΞ(dν) has law Qτ,m (Ht ∈ ·). with intensity Rτ,t (y, ·)dm(y) then The fact that wτ = y for Rτ,t (y)-a.a. w (in (a)) is immediate from (c) and the corresponding property for Ht under Qτ,δy (use (II.8.5) to see the latter). The uniqueness of the canonical measure in Theorem II.7.1 and the fact that under Qτ,δy , Xt (·) = Hτ +t ({y  : yτ +t ∈ ·}) is a (Y, γ, 0)-superprocess starting at ˆ t (y) = yt and (Rt (x))x∈E are the yτ (y ∈ Dτ ) by Exercise II.8.3, show that if Π canonical measures for X, then (II.8.7)

ˆ −1 ∈ ·) = Rt (yτ , ·). Rτ,τ +t (y, ν ◦ Π τ +t

III. Sample Path Properties of Superprocesses 1. Historical Paths and a Modulus of Continuity  with γ(·) ≡ γ > 0 Assume X, (Pµ )µ∈MF (E) is a (Y, γ, 0)-DW-superprocess  constant, (PC) holds, and H, (Qτ,m )(τ,m)∈Eˆ is the corresponding historical process on their canonical path spaces, ΩX and ΩH , respectively. Theorem III.1.1. (Historical Cluster Representation). Let m ∈ MFτ (D) and τ ≤ s < t. If rs (Ht )(·) = Ht ({y : y s ∈ ·}), then M   D Qτ,m rs (Ht ) ∈ · | F H [τ, s+] (ω)≡ ei δyi ,

(III.1.1)

i=1

where (ei , yi )i≤M are the  points of a Poisson point process on R+ × D with intensity 2 ) νt−s × Hs (ω) and νt−s is an exponential law with mean γ(t − s)/2. That ( γ(t−s) is, the right-hand side of (III.1.1) defines a regular conditional distribution for the random measure on the left side given F H [τ, s+]. Proof. By the Markov property of H we may assume s = τ . Fix m ∈ MFτ (D). Let A1 , . . . , An be a Borel partition of D and define mi = m(· ∩ Ai ). Let H 1 , . . . , H n be independent (Ft )-historical processes with H i starting at (τ, mi) on some (Ω, F , Ft, P). Then by checking the historical martingale problem (HM P )τ,m we can easily see n H i is an (Ft )-historical process starting at (τ, m) and so has law Qτ,m . that H ≡ i=1

For each i, the mean value property for historical processes (II.8.5) implies (recall y = y τ m-a.e.) for each t > τ   P Hti ({y : y τ ∈ Aci }) = 1Ai (y τ )Pτ,yτ (y τ ∈ Aci )dm(y) = 0. The process inside the expected value on the left side is a.s. continuous in t by Exercise II.8.2 and so is identically 0 for all t ≥ τ a.s. This implies (III.1.2)

Hti (·) = Ht (· ∩ {y τ ∈ Ai })

i = 1 . . . n for all t ≥ τ

a.s.

It follows that if we start with H· under Qτ,m and define H i by (III.1.2), then (III.1.3)

(H·i , i ≤ n) are independent F H [τ, t+]-historical processes starting at (τ, mi )i≤n , respectively.

i (1), s ≥ 0}i=1...n are independent Feller diffusions (i.e. solutions In particular {Hs+τ of (II.5.10) with g = 0) with initial values {m(Ai ) : i = 1 . . . n}, respectively. Recalling the Laplace transforms of Hti (1) from (II.5.11), we have for any λ1 , . . . , λn ≥ 0

194 and f (y) =

Superprocesses n

λi 1Ai (y) (y ∈ Dτ ),

i=1 n        = Qτ,m exp − λi Hti (1) Qτ,m exp − f (y τ )Ht (dy)

 = exp 

i=1

−

n  i=1



2λi m(Ai )  2 + λi (t − τ )γ

2f (y)(2 + f (y)(t − τ )γ)−1 dm(y)    = E exp − ef (y)Ξ(de, dy) , = exp

−



  2 where Ξ is a Poisson point process on R+ × D with intensity γ(t−τ νt−τ × m . ) As the above equation immediately
follows for any Borel
f ≥ 0 on D, the proof is complete because we have shown 1(y τ ∈ ·)Ht (dy) and e1(y ∈ ·)Ξ(de, dy) have the same Laplace functionals. A consequence of the above argument is Lemma If A is a Borel subset of D(E), and m ∈ MFτ (D), then Xs =  III.1.2. τ Hs+τ {y : y ∈ A} is a Feller branching diffusion (a solution of (II.5.10) with g = 0), and for t > τ , * + m(A)2 Qτ,m (Hs ({y : y τ ∈ A}) = 0 ∀s ≥ t) = exp − . γ(t − τ ) Proof. This is immediate from (III.1.3) and the extinction probabilities found in (II.5.12).   The above Theorem shows for s < t, Ht {y : y s ∈ ·} is a purely atomic measure. The reader should be able to see that (conditionally) this measure is a Poisson superposition of exponential masses directly from Kolmogorov’s and Yaglom’s theorems (Theorem II.1.1). If τ = s = 0, the above may also be easily derived from the corresponding canonical measure representation (II.7.11) for Ht and projecting it down to y0 . Note that the exponential masses come from the last assertion of Theorem II.7.2(iii). An extension of the above cluster decomposition which describes the future evolution of the descendants of these finite number of clusters will be proved using similar ideas in Section III.6 (see Theorem III.6.1 and Corollary III.6.2).  Until otherwise indicated assume (Xt )t≥0 , (Pµ )µ∈MF (Rd ) is a super-Brownian motion with branching rate γ > 0 (we write X is SBM( γ)). This means X is a (B, γ, 0)-DW-superprocess on its canonical space ΩX , where B is a standard Brownian motion in Rd and γ is a positive constant. H· will denote the corresponding historical process on ΩH . We call H a historical Brownian motion (with branching rate γ). The following result is essentially proved in Dawson-Iscoe-Perkins (1989) but first appeared in this form as Theorem 8.7 of Dawson-Perkins (1991). It gives a uniform modulus of continuity for all the paths in the closed support of Ht for all t ≥ 0. The simple probabilistic proof given below seems to apply in a number of different settings. See, for example, Mueller and Perkins (1992) where it is applied to

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195

the supports of the solutions of a class of parabolic stochastic pde’s. It also extends readily to more general branching mechanisms (see Dawson and Vinogradov (1994)) and to the interacting models considered in Chapter V (see Chapter 3 of Perkins (1995)). Notation: S(µ) denotes the closed support of a measure µ on the Borel sets of a  1/2 is L´evy’s modulus function. metric space. h(r) = r log 1/r Theorem III.1.3. (Historical Modulus of Continuity). If δ, c > 0, let K(δ, c) = {y ∈ C(Rd ) : |y(r) − y(s)| ≤ ch(|r − s|) ∀r, s ≥ 0 satisfying |r − s| ≤ δ}. Q0,m denotes the law of historical Brownian motion with branching rate γ > 0 starting at (0, m). (a) If c > 2, then Q0,m -a.s. there is a δ(c, ω) > 0 such that   S(Ht )(ω) ⊂ K δ(c, ω), c ∀t ≥ 0. Moreover there are constants ρ(c) > 0 and C(d, c) such that Q0,m (δ(c) ≤ r) ≤ C(d, c)m(1)γ −1 r ρ for r ∈ [0, 1]. (b) If m  = 0 and  c < 2, then Q0,m -a.s. for all δ > 0 there is a t in (0, 1] such that Ht K(δ, c)c > 0. Remark. This should be compared to L´evy’s modulus of continuity for a simple √ Brownian path for which c = 2 is critical. This reflects the fact that the tree of Brownian paths underlying H· has infinite length. We prove (a) below only for a sufficiently large c. Proof of (a) (for large c). Use Lemma III.1.2 and the Markov property to see that  j − 1       j    −n n ) > 0 ∃t ≥ (j + 1)/2 > ch(2 Q0,m Ht y : y n ) − y  2 2n   2n+1   j  j − 1     Hj/2n y : y( n ) − y = Q0,m 1 − exp −  > ch(2−n ) n γ 2 2    ≤ 2n+1 γ −1 Q0,m Hj/2n |y(j/2n ) − y((j − 1)/2n )| > ch(2−n ) . Now recall from (II.8.5) that the mean measure of Ht is just Wiener measure stopped at t. The above therefore equals   2n+1 γ −1 Py |B(j/2n ) − B((j − 1)/2n )| > ch(2−n ) dm(y) ≤ 2n+1 γ −1 m(1)cd nd/2−1 2−nc

2

/2

by a Gaussian tail estimate. Sum over 1 ≤ j ≤ n2n to see that      j   j − 1    > 0, Q0,m Ht y : y n − y  > ch(2−n ) n 2 2 for some t ≥ (j + 1)2−n and 1 ≤ j ≤ n2n ≤ cd γ −1 m(1)nd/2 22n+1 2−nc

2

/2

,



196

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which is summable if c > 2. Assuming the latter, we may use Borel-Cantelli to see ∃N (ω) < ∞ a.s. such that Ht ≡ 0 for t ≥ N (ω) and  j  j − 1    n ≥ N (ω) ⇒ y n ) − y  ≤ ch(2−n ) ∀j ≥ 1, (j + 1)2−n ≤ t, 2 2n (III.1.4) Ht -a.a. y ∀t ≤ n. We now follow L´evy’s proof for Brownian motion. Let δ(c2 , ω) = 2−N(ω) > 0 a.s., The required bound on where c2 will be chosen large enough below. Q0,m (δ(c2 ) ≤ r) is clear from the above bound and the extinction probability estimate formula (II.5.12). Let N ≥ t > 0 and choose y outside of an Ht -null set so that (III.1.4) holds. Assume r < s ≤ t and 0 < s − r ≤ 2−N and choose n ≥ N so that 2−n−1 < s − r ≤ 2−n . For k ≥ n, choose sk ∈ {j2−k : j ∈ Z+ } such that sk + 2−k ≤ s and sk is the largest such value (set sk = 0 if s < 2−k ). One easily checks that sk ↑ s, sk+1 = sk + jk+1 2−(k+1) for jk+1 = 0, 1 or 2 (and jk+1 = 0 only arises if s < 2−k−1 ). Note also that sk + (jk+1 + 1)2−k−1 = sk+1 + 2−k−1 ≤ s ≤ t. Therefore the choice of y and (III.1.4) imply |y(sk+1 ) − y(sk )| ≤ jk+1 ch(2−k−1 ), and so for some c1 > 0 (c1 = 5c/ log 2 will do),

(III.1.5)

|y(s) − y(sn )| ≤

∞ 

jk+1 ch(2−k−1 ) ≤ 2c

k=n

∞ 

h(2−k−1 ) ≤ c1 h(2−n−1 )

k=n

≤ c1 h(s − r). Similarly one constructs rk ↑ r so that (III.1.6)

|y(r) − y(rn )| ≤ c1 h(s − r).

The restriction s − r ≤ 2−n implies sn = rn + jn 2−n where jn = 0 or 1 and so sn ≤ s ≤ t, which means by (III.1.4) that √ (III.1.7) |y(sn ) − y(rn )| ≤ ch(2−n ) ≤ c 2h(s − r). √ (III.1.5)-(III.1.7) imply |y(s) − y(r)| ≤ (2c1 + c 2)h(s − r) ≡ c2 h(s − r). This proves   Ht K(δ(c2 ), c2 )c = 0 for t ≤ N (ω) and so  all t ≥  0 because Ht = 0 for t > N . K(δ, c2 ) is closed and therefore S(Ht ) ⊂ K δ(c2 ), c for all t ≥ 0. To get (a) for any c > 2, one works with a finer set of grid points {(j + p n M )θ : j ∈ Z+ , 0 ≤ p < M } (θ < 1 sufficiently small and M large) in place of {j2−n : j = 0, 1, 2 . . .} to get a better approximation to r, s, as in L´evy’s proof for Brownian motion. For example, see Theorem 8.4 of Dawson-Perkins (1991). (b) Let c < 2 and 1 > η > 0. If Bjn = {y ∈ C : |y(2j2−n ) − y((2j − 1)2−n )| > ch(2−n )},

Superprocesses

197

it suffices to show (III.1.8) H(2j+1)2−n (Bjn ) > 0 for large n for a.a. ω satisfying inf Ht (1) ≥ η. sup t<η

j∈N,(2j+1)2−n <η

This is because the first event implies supt<η Ht (K(δ, c)c ) > 0 ∀δ > 0, and Q0,m (inf t<η Ht (1) ≥ η) ↑ 1 as η ↓ 0. If Anj = {ω : H(2j+1)2−n (Bjn ) = 0, H(2j+1)2−n (1) ≥ η}, then we claim that ∞ 

(III.1.9)

Q0,m (∩1≤j,(2j+1)2−n ≤η Anj ) < ∞.

n=1

Assume this for the moment. The Borel-Cantelli Lemma implies w.p.1 for n large enough ∃(2j + 1)2−n < η such that H(2j+1)2−n (Bjn ) > 0 or H(2j+1)2−n (1) < η, which implies w.p.1 either inf Ht (1) < η or for large n t<η

sup

(2j+1)2−n <η

H(2j+1)2−n (Bjn ) > 0.

This gives (III.1.8) and so completes the proof. Turning to (III.1.9), note that Q0,m (Anj |F H [0, 2j2−n +]) ≤ Q0,m (H(2j+1)2−n (Bjn ) = 0|F H [0, 2j2−n+])   ≤ exp −2n+1 γ −1 H2j2−n (Bjn ) , the last by Theorem III.1.1 with t = (2j +1)2−n and s = 2j2−n . Let Rs,t (y, ·) be the canonical measures associated with H, introduced in (II.8.6). Condition the above with respect to F H [0, (2j − 1)2−n +] and use (II.8.6)(b) and the Markov property to see that Q0,m (Anj |F H [0, (2j − 1)2−n +])   ≤ exp − 1 − exp(−2n+1 γ −1 ν(Bjn ))R(2j−1)2−n ,2j2−n (y, dν)H(2j−1)2−n (dy)  ≤ exp − 1(ν(1) ≤ 2−n )[1 − exp(−2n+1 γ −1 ν(Bjn ))]  R(2j−1)2−n ,2j2−n (y, dν)H(2j−1)2−n (dy)  ≤ exp −c0 1(ν(1) ≤ 2−n )2n+1 γ −1 ν(Bjn )  R(2j−1)2−n ,2j2−n (y, dν)H(2j−1)2−n (dy) ,

198

Superprocesses

where c0 = c0 (γ) > 0 satisfies 1 − e−2x/γ ≥ c0 x for all x ∈ [0, 1]. (II.8.6)(a) shows that on Anj−1 Q0,m (Anj |F H [0, (2j − 1)2−n +]) ∞  1(γ2−n−1 z ≤ 2−n )ze−z dz ≤ exp −c0 2n+1 γ −1 0  × P(2j−1)2−n ,y (|B(2j2−n ) − B((2j − 1)2−n | > ch(2−n ))H(2j−1)2−n (dy)  

≤ exp −c0 2n+1 γ −1 c1 P 0 (|B(1)| ≥ c n log 2)H(2j−1)2−n (1) ≤ exp{−c2 (γ, c)η2n(1−c

2

/2) −1/2

n

},

for some c2 > 0, at least for n ≥ n0 (c) by a Gaussian tail estimate and the fact that H(2j−1)2−n (1) ≤ η on Anj−1 . Therefore for n ≥ n1 (c, η), 

Q0,m ∩j≥1,(2j+1)2−n ≤η Anj



 −c η 2 2n(2−c2 /2)  2 √ ≤ exp , 2 n

which is summable over n since c < 2. This gives (III.1.9) and we are done. Remark. It is easy to obtain versions of the above result for continuous spatial motions other than Brownian motion (see Theorem 8.6 in Dawson-Perkins (1991)). Notation. Πt : D(E) → E is the projection map Πt (y) = y(t). Recall X denotes SBM(γ) and H is the associated historical Brownian motion. Corollary III.1.4. (a) S(Ht ) is compact in C(Rd ) ∀t > 0 Q0,m -a.s.  (b) S(Xt ) = Πt S(Ht ) and hence is compact in Rd ∀t > 0 Pm -a.s. Proof. (a) Lemma III.1.2 shows that for any η > 0, 

Q0,m Hs







*

2m({(y0 ) > R}) {|y0 | > R} = 0 for all s ≥ η = exp − γη

+

→ 1 as R → ∞. This and the previous theorem show that for Q0,m -a.a. ω there is a δ(3, ω) > 0 and an R(ω) < ∞ such that   S(Ht )(ω) ⊂ K δ(3, ω), 3 ∩ {y : |y0 | ≤ R(ω)} ∀t ≥ η. The set on the righthand side is compact by the Arzela-Ascoli theorem. Let η ↓ 0 to complete the proof of (a). −1 for all t ≥ 0. Note that (b) From (II.8.4)  we may assume that Xt = Ht ◦ Πt  −1 S(Ht ) ⊂ Πt Πt S(Ht ) and therefore        c  c  = Ht Π−1 Πt S(Ht ) ≤ Ht S(Ht )c = 0. Xt Πt S(Ht ) t

Superprocesses

199

  This  shows  > 0 by (a), S(Xt ) ⊂  Πt S(Ht ) supports Xt and as it is compact ∀t Πt S(Ht ) ∀t > 0 and is also compact a.s. If w ∈ S(Ht ), Ht {y : |yt − wt | < ε} > 0 for all ε > 0 and so wt ∈ S(Xt ). This shows the reverse inclusion. A measure-valued process X· has the compact support property (CSP) iff S(X0 ) compact implies S(Xt ) is compact for all t > 0 a.s. Corollary III.1.4 shows that super-Brownian motion has the (CSP) (in fact S(X0 ) need not be compact). Obviously this property fails for the heat kernel Pt φ. The (CSP) for SBM was first proved by Iscoe (1988). The next result provides the natural rate of propagation for X suggested by the historical modulus of continuity. Notation. A ⊂ Rd , δ > 0, Aδ = {x ∈ Rd : d(A, x) ≡ inf{|y − x| : y ∈ A} < δ}. Corollary III.1.5. With probability 1 for any c > 2 there is a δ(c, ω) > 0 such that if 0 < t − s < δ(c, ω), then S(Xt ) ⊂ S(Xs )ch(t−s) . To avoid an unexpected decline in S(Xs ) on the right side of this inclusion we need a lemma. Lemma III.1.6. For Q0,m -a.a. ω if 0 ≤ s ≤ t and y ∈ S(Ht ), then y(s) ∈ S(Xs ). Proof. If 0 ≤ s < s are fixed, Theorem III.1.1 shows that conditional on F H [0, s+],  Hs y(s) ∈ · is supported on a finite number of points x1 . . . xn in S(Xs ). The Markov property and (III.1.3) show that conditional on F H [0, s +], {Ht (ys = xi ) : t ≥ s }i≤n and {Ht (ys ∈ {x1 . . . xn }) : t ≥ s } are independent Feller diffusions. The latter is therefore a.s. identically 0 and so w.p.1 for all t ≥ s ,   (trivial) {y(s) : y ∈ S(Ht )} ⊂ S Ht (ys ∈ ·)   ⊂ S Hs (ys ∈ ·) ⊂ S(Xs ). Take the union over all 0 ≤ s ≤ s in Q to conclude (III.1.8) w.p.1 for all s ∈ Q≥0

and all t > s {y(s) : y ∈ S(Ht )} ⊂ S(Xs ).

A simple consequence of our modulus of continuity and Xt = Ht (yt ∈ ·) is that if B = B(x, ε), B  = B(x, ε/2) and m(B) = 0, then Qτ,m -a.s. ∃η > 0 such that Xt (B  ) = 0 for all τ ≤ t < η. Use this and the strong Markov property at time Tr (B) = inf{t ≥ r : Xt (B) = 0} where B = B(x, ε) is a rational ball (x ∈ Qd , ε ∈ Q>0 ) and B  is as above to conclude: (III.1.9)

w.p.1 for all r ∈ Q≥0 and rational ball B ∃η > 0 such that Xs (B  ) = 0 for all s in [Tr (B), Tr (B) + η).

Choose ω so that (III.1.8) and (III.1.9) hold. Let y ∈ S(Ht ), s < t and suppose y(s) ∈ S(Xs ) (the s = t case is handled by Corollary III.1.4). Choose a rational ball B so that y(s) ∈ B  and Xs (B) = 0, η  > 0, and a rational r in (s − η  , s]. Then

200

Superprocesses

Tr (B) ≤ s because Xs (B) = 0 and so by (III.1.9) there is an open interval I in (s − η  , s + η  ) such that Xu (B  ) = 0 for all u in I. In particular there are rationals un → s such that Xun (B  ) = 0. On the other hand by (III.1.8) and the continuity of y, y(un ) ∈ B  and y(un ) ∈ S(Xun ) for n large which implies Xun (B  ) > 0, a contradiction. Proof of Corollary III.1.5. Apply Theorem III.1.3, Corollary III.1.4 and Lemma III.1.6 to see that w.p.1 if 0 < t − s < δ(c, ω),    ch(t−s) ⊂ S(Xs )ch(t−s) . S(Xt ) = Πt S(Ht ) ⊂ Πs S(Ht ) Remark. Presumably c = 2 is also sharp in Corollary III.1.5 if d ≥ 2, although this appears to be open. It would be of particular interest to find the best result in d = 1 as the behaviour of ∂S(Xt) in d = 1 could shed some light on the SPDE for super-Brownian motion in one dimension. Definition. For I ⊂ R+ we call R(I) =



S(Xt ), the range of X on I, and    R(I) = R(I) is the closed range of X on I. The range of X is R = R [δ, ∞) . t∈I

δ>0

It is not hard to see that R − R(0, ∞) is at most a countable set of “local extinction points” (see Proposition 4.7 of Perkins (1990) and the discussion in Section III.7 below). R  easierto deal with than R((0, ∞)). The reason for not  is sometimes  considering R [0, ∞) or R [0, ∞) is that it will be Rd whenever S(X0 ) = Rd .     Corollary III.1.7. R [δ, ∞) is compact for all δ > 0 a.s. R [0, ∞) is a.s. compact if S(X0 ) is. Proof. Immediate from Corollaries III.1.4 and III.1.5. In view of the increase in the critical value of c in Theorem III.1.3 from that for a single Brownian path, it is not surprising that there are diffusions Y for which the (CSP) fails for the associated DW-superprocess X. Example 8.16 in DawsonPerkins (1991) gives a time-inhomogeneous R-valued diffusion and T > 0 for which S(XT ) = φ or R a.s. For jump processes instantaneous propagation is to be expected as is shown in the next Section. 2. Support Propagation and Super-L´ evy Processes Let Y be a Poisson (rate λ > 0) process on Z+ and consider X, the  (Y, γ, 0)DW superprocess with γ > 0 constant. Then Aφ(i) = λ φ(i + 1) − φ(i) , i ∈ Z+ , and taking φ(i) = 1(i = j) in the martingale problem for X, we see that Xt =   Xt (j) j∈Z+ may also be characterized as the pathwise unique solution of t (III.2.1) Xt (j) = X0 (j) + λ 0

  Xs (j − 1) − Xs (j) ds +

t γXs (j)dBsj , j ∈ Z+ . 0

Here {B·j : j ∈ Z+ } is a collection of independent linear Brownian motions, and Xs (−1) ≡ 0. Pathwise uniqueness holds by the method of Yamada and Watanabe (1971).

Superprocesses

201

  Let X0 = αδ0 . Note that Xt {0, . . . , n} is a non-negative supermartingale and so sticks at 0 when it first hits 0 at time ζn . Evidently ζn ↑ ζ = inf{t : Xt (Z+ ) = 0} < ∞ a.s. Clearly mt = inf S(Xt ) = n if t ∈ [ζn−1 , ζn ) (ζ−1 = 0) and so is increasing in t a.s., and Xt becomes extinct as the lower end of its support approaches infinity. On the other hand it is clear (at least intuitively) from (III.2.1) that the mass at mt will immediately propagate to mt + 1 which in turn immediately propagates to mt + 2, and so on. Therefore we have S(Xt ) = {mt , mt + 1, . . .} a.s. ∀t > 0.

(III.2.2)

This result will be a special case of Theorem III.2.4 below. Note, however, that S(Xt ) = {mt } at exceptional times is a possibility by a simple comparison argument with the square of a low-dimensional Bessel process (see Exercise III.2.1). The above example suggests S(Xt ) will propagate instantaneously to any points to which Y can jump. This holds quite generally (Corollary 5.3 of Evans-Perkins (1991)) but for the most part we consider only d-dimensional L´evy processes here. Our first result, however, holds for a general (Y, γ, g)-DW superprocess X with law PX0 . Recall if Zt = Z0 + Mt + Vt is the canonical decomposition of a semimartingale then its local time at 0 is (LT )

L0t (Z)

−1

t 1(0 ≤ Zs ≤ ε)dM s

= lim ε ε↓0

a.s.

0

and in particular ∞ 1(Zs = 0)dM s = 0.

(III.2.3) 0

Theorem III.2.1. If φ ∈ D(A)+ , then with probability 1 for Lebesgue-a.a. s, Xs (Aφ) > 0 implies Xs (φ) > 0. Proof. This should be intuitively clear from (M P )X0 as the drift Xs (Aφ)ds > 0
t should keep Xs (φ) > 0 a.s. Since Xt (φ) = X0 (φ) + Mt (φ) + Xs (Ag φ)ds, we have 0

with probability one,

202

Superprocesses

Lt ≡

L0t



t



X(φ) = lim ε↓0

  1 0 < Xs (φ) ≤ ε Xs (γφ2 )dsε−1

0

t ≤ γφ lim ε↓0

  1 0 < Xs (φ) ≤ ε Xs (φ)ε−1 ds

0

t ≤ γφ lim ε↓0

  1 0 < Xs (φ) ≤ ε ds

0

=0

a.s.

Tanaka’s Formula implies t Xt (φ)+ = X0 (φ) +

t 1 (Xs (φ) > 0) dMs (φ) +

0

1 (Xs (φ) > 0) Xs (Aφ)ds 0

t 1 (Xs (φ) > 0) Xs (gφ)ds.

+ 0

Clearly


t

1(Xs (φ) = 0) Xs (gφ) ds = 0, and by (III.2.3),

0


t  1 Xs (φ) = 0 dMs (φ) = 0. 0

The above therefore implies t

t Xs (A φ)ds − g

+

Xt (φ) = Xt (φ) = X0 (φ) + Mt (φ) + 0

1(Xs (φ) = 0)Xs (Aφ)ds 0

t = Xt (φ) −

1(Xs (φ) = 0)Xs (Aφ)ds. 0

We conclude that


t

1(Xs (φ) = 0)Xs (Aφ)ds = 0 ∀t ≥ 0 a.s. and the result fol-

0

lows. Assume now that Y is a L´evy process in Rd with L´evy measure ν. Then D(A) ∞ contains CK , the C ∞ -functions on Rd with compact support. Let B be an open ∞ )+ such that {φ > 0} = B. Then for x ∈ B, ball in Rd
and choose φ ∈ (CK Aφ(x) = φ(x + y)ν(dy) (see, e.g., Theorem IV.4.1 of Gihman and Skorokhod (1975), or Example II.2.4(b) when Y is an asymmetric α-stable process). This means that Xs (B) = 0 implies Xs (Aφ) = Xs ∗ ν(φ), where ∗ denotes convolution of measures. Theorem III.2.1 therefore implies w.p.1 Xs ∗ν(B) > 0 implies Xs (B) > 0, for Lebesgue a.a. s. Taking a union over balls with rational radii and centers we conclude (III.2.4)

S(Xs ∗ ν) ⊂ S(Xs )

for Lebesgue a.a. s > 0 a.s.

Superprocesses

203

The “Lebesgue a.a. s” is a nuisance as we would like to verify this inclusion for a fixed s > 0 (it is false for all s > 0 simultaneously as Exercise III.2.1 shows). The following result allows us to do this and also has several other applications. Theorem III.2.2. Let X be the (Y, γ, g)-DW-superprocess where γ(·) ≡ γ > 0 is constant and g ∈ Cb (E). Let µ1 , µ2 ∈ MF (E). The following are equivalent: (i) µ1 Ps  µ2 Pt ∀ 0 < s ≤ t (ii) Pµ1 (Xs ∈ ·)  Pµ2 (Xt ∈ ·) ∀ 0 < s ≤ t (iii) Pµ1 (Xs+· ∈ ·)  Pµ2 (Xt+· ∈ ·) (on C(R+ , MF (E))) ∀0 < s ≤ t. The original proof in Evans-Perkins (1991) used exact moment measure calculations and Theorem II.7.2. A simpler argument using only the latter is given at the end of the Section. Example III.2.3. Let X be a super-α-stable process, i.e. g = 0, γ(·) = γ > 0 constant and Y is the symmetric α-stable process in Example II.2.4(b) (and so is Brownian motion if α = 2). For any µ1 , µ2 ∈ MF (Rd ) − {0}, (i) is trivial as µ1 Ps is equivalent to Lebesgue measure for all s > 0. Therefore Pµ1 (Xs ∈ ·) and Pµ2 (Xt ∈ ·) are equivalent measures on MF (Rd ) and Pµ1 (X ∈ ·) and Pµ2 (X ∈ ·) are equivalent measures on (ΩX , σ(Xr : r ≥ δ)) for any δ > 0. For 0 < α < 2 the first equivalence allows us to consider a fixed s in (III.2.4) and conclude S(Xs ∗ ν) ⊂ S(Xs ) a.s. ∀s > 0. Recall that ν(dx) = c|x|−d−α dx and conclude (III.2.5)

S(Xs ) = φ

or

Rd

a.s. ∀s > 0.

A similar application of Theorem III.2.2 easily gives (III.2.2) for super-Poisson processes. More generally we have Theorem III.2.4. Let Y be a L´evy process on Rd with L´evy measure ν, let γ(·) ≡ γ > 0 be constant and let X be the (Y, γ, 0)-DW-superprocess starting at X0 under PX0 . If ν ∗k is the k-fold convolution of ν with itself then ∞ 

S(ν ∗k ∗ Xt ) ⊂ S(Xt ) PX0 − a.s. ∀t > 0,

X0 ∈ MF (Rd ).

k=1

Proof. Choose X0 in MF (Rd ) so that X0 (A) = 0 iff A is Lebesgue null. Then P X0 (Yt ∈ A) = 0 iff A is Lebesgue null and so just as in the α-stable case above, Theorem III.2.2 and (III.2.4) imply that if Λ = {µ ∈ MF (E) : S(ν ∗ µ) ⊂ S(µ)} then PX0 (Xt ∈ Λ) = 1 ∀t > 0. The cluster decomposition (II.7.11) implies that for each t > 0, Rt (x0 , Λc ) = 0

(III.2.6)

for Lebesgue a.a. x0 .


Let τy : MF (Rd ) → MF (Rd ) be the translation map τy (µ)(A) = 1A (x + y)dµ(y). Then Rt (x0 , τy−1 (·)) = Rt (x0 + y, ·), e.g., by Theorem II.7.2(i) and the translation invariance of XtN . Clearly τy−1 (Λ) = Λ and so (III.2.6) implies Rt (x0 , Λc ) = 0 for any x0 ∈ Rd . Another application of the cluster decomposition (II.7.11) (use n n  S(µi ) = S( µi )) shows S(ν ∗ Xt ) ⊂ S(Xt ) PX0 -a.s. for any X0 ∈ MF (Rd ). i=1

1

Iterate this to complete the proof.

204

Superprocesses

Remark. Dawson’s Girsanov theorem (Theorem IV.1.6 below) immediately gives the above for a general non-zero drift g in Cb (E). It is interesting to compare Theorem III.2.4 with the following result of Tribe (1992). Theorem III.2.5. Assume Y is a Feller process on a locally compact metric space E, X is the Y -DW-superprocess starting at X0 ∈ MF (E) under PX0 , and ζ = inf{t : Xt (1) = 0}. Then there is a random point F in E such that   PX0 F ∈ A | Xs (1), s ≥ 0 (ω) = P X0 (Yζ(ω) ∈ A)/X0 (1) a.s. ∀A ∈ E and lim t↑ζ

Xt (·) = δF Xt (1)

a.s. in MF (E).

Proof. See Exercise III.2.2 below. This says the final extinction occurs at a single “death point” F , which, in view of the independence assumptions underlying X, is to be expected. On the other hand Example IV.2.3 shows that for an α-stable superprocess S(Xt ) = Rd for Lebesgue a.a. t < ζ a.s. because of the ability of the mass concentrating near F to propagate instantaneously. The following result of Tribe (1992) shows the support will collapse down to F at exceptional times at least for α < 1/2. Theorem III.2.6. Assume X is a α-stable-DW-superprocess with α < 1/2. Let F be as in Theorem III.2.5. For a.a. ω there are sequences εn ↓ 0 and tn ↑ ζ such that S(Xtn ) ⊂ B(F, εn ). We close this Section with the Proof of Theorem III.2.2.  The  implications (ii) ⇒ (i) and (ii) ⇒ (iii) are immediate by considering Pµ1 Xs (·) and using the Markov property, respectively. As (iii) ⇒ (ii) is trivial, only (i) ⇒ (ii) requires a proof. Dawson’s Girsanov Theorem (Theorem IV.1.6 below) reduces this implication to the case where g ≡ 0 which we now assume.
Assume (i) and choose 0 < s ≤ t. Write Ru (m, ·) for Ru (x0 , ·)dm(x0 ) and set Ru∗ (m, ·) = Ru (m, ·)/Ru(m, MF (E) − {0}), where Ru (x0 , ·) are the canonical measures of X from Theorem II.7.2 and m ∈ MF (E) − {0}. The first step is to reduce the problem to Rs (µ1 , ·)  Rt (µ2 , ·) on MF (E) − {0}. N1 1 N2 2 By (II.7.11), Pµ1 (Xs ∈ ·) and Pµ2 (Xt ∈ ·) are the laws of i=1 νi and i=1 νi , respectively, where N1 and N2 are Poisson with means 2µ1 (1)/γs and 2µ2 (1)/γt, respectively, and conditional on N1 , N2 , {νi1 : i ≤ N1 } and {νi2 : i ≤ N2 } are i.i.d. with law Rs∗ (µ1 , ·) and Rt∗ (µ2 , ·), respectively. (III.2.7) implies the n-fold product of Rs∗ (µ1 , ·) will be absolutely continuous to the n-fold product of Rt∗ (µ2 , ·). Therefore we can sum over the values of N1 and N2 to obtain Pµ1 (Xs ∈ ·)  Pµ2 (Xt ∈ ·) as required. Let Ξr,ν0 denote a Poisson point process on MF (E)−{0} with intensity Rr (ν0 , ·). If 0 < τ < t, then Exercise II.7.2 (b) and (II.7.11) show that (III.2.8)   ν Ξt−τ,ν0 (dν) ∈ · Rτ (µ2 , dν0 ) as measures on MF (E) − {0}. Rt (µ2 , ·) = P (III.2.7)

Superprocesses

205

2ν0 (1) This and the fact that Rt−τ (ν0 , 1) = γ(t−τ ) , show that   ν Ξt−τ,ν0 (dν) ∈ ·, Ξt−τ,ν0 (MF (E) − {0}) = 1) Rτ (µ2 , dν0 ) Rt (µ2 , ·) ≥ P  2ν (1)  0 (III.2.9) Rt−τ (ν0 , ·) Rτ (µ2 , dν0 ). = exp − γ(t − τ )

Assume now that B is a Borel subset of MF (E) − {0} such that Rt (µ2 , B) = 0. Then (III.2.9) implies that 0= Rt−τ (x0 , B)dν0 (x0 ) Rτ (µ2 , dν0 ). Recall from Theorem II.7.2 (c) that the mean measure associated with Rτ (µ2 , ·) is µ2 Pτ . Therefore the above implies that Rt−τ (x0 , B) = 0

µ2 Pτ − a.a. x0 .

Now apply (i) to see that for 0 ≤ h < τ , Rt−τ (x0 , B) = 0 µ1 Pτ −h − a.a. x0 . Now reverse the above steps to conclude 0 = Rt−τ (ν0 , B)Rτ −h (µ1 , dν0 ), and for s > ε > 0, set τ = t − s + ε and h = t − s to get Rs−ε (ν0 , B)Rε (µ1 , dν0 ) = 0. Use this result in (III.2.9) with our new parameter values to see that   0= P ν Ξs−ε,ν0 (dν) ∈ B, Ξs−ε,ν0 (MF (E) − {0}) = 1 Rε (µ1 , dν0 )   (III.2.10) ≥ Rs (µ1 , B) − P Ξs−ε,ν0 (MF (E) − {0}) ≥ 2 Rε (µ1 , dν0 ), where in the last line we used (III.2.8) with our new parameter values and also the fact that 0 ∈ / B so that there is at least one point in the Poisson point process Ξs−ε,ν0 . The elementary inequality 1 − e−x − xe−x ≤ x2 /2 for x ≥ 0 and Theorem II.7.2 (iii) show that if γ  = γ(s − ε)/2, then the last term in (III.2.10) is   1 − exp(−ν0 (1)/γ  ) − (ν0 (1)/γ  ) exp(−ν0 (1)/γ  ) Rε (µ1 , dν0 ) ∞  2 2   x e−2x/γε dx = 1 − e−x/γ −  e−x/γ γ γε 0 ∞ 2  2 x 2 e2x/γε dx ≤ 2 γε 2γ 0 = γ −2 γε/2.

206

Superprocesses

Use this bound in (III.2.10) to conclude that Rs (µ1 , B) ≤ γ −2 γε/2 and for any ε as above and hence Rs (µ1 , B) = 0, as required. Exercise III.2.1. Let Xt be the super-Poisson process which satisfies (III.2.1)  with X0 = αδ0 (α > 0). Prove that P S(Xt ) = {0} ∃t > 0 > 0 and conclude that (III.2.2) is not valid for all t > 0 a.s. Hints. By a simple scaling argument we may take γ = 4. (i) Show that t Xt (0) = α −

t λXs (0)ds + 2 Xs (0)dBs0

0

  Xt ≡ Xt {0}c = 0

0

t

t λXs (0)ds + 2 Xs dBs ,

0

0



where (B , B ) are independent linear Brownian motions.   (ii) Let T1 = inf{t : Xt (0) ≤ λ−1 }, T2 = inf{t : t > T1 , Xt (0) ∈ 12 λ−1 , 32 λ−1 } − T1 and let t+T 1 3 2 Ys dBs . Yt = XT 1 + t + 2 T1

Y is the square of a 3/2-dimensional Bessel process and hits 0 a.s. (see V.48 of Rogers and Williams (1987)), Y· and X·+T1 (0) are conditionally independent given FT1 . Argue that Y hits 0 before T2 with positive probability and use this to infer the result. Exercise III.2.2. (Tribe (1992)). Let X be a Y -DW-superprocess (hence g ≡ 0, γ constant) on (Ω, F , Ft, P), where Y is a Feller process on a locally compact metric space E and X0 ∈ MF (E) − {0}. This means Pt is norm continuous on C (E) the space of bounded continuous functions with a finite limit at ∞. Let E∞ = E ∪ {∞} be the one-point compactification of E.
t (a) Let Ct = Xs1(1) ds for t < ζ. It is well-known and easy to show that Cζ− = ∞ 0

(see Shiga (1990), Theorem 2.1). Let Dt = inf{s : Cs > t} for t ≥ 0, Z˜t (·) = XDt (·) and Zt (·) = Z˜t (·)/Z˜t (1). Show that if φ ∈ D(A), then t Z˜s (Aφ)ds + Nt (φ),

Zt (φ) = X0 (φ)/X0 (1) + 0

where Nt (φ) is a continuous (FDt )-martingale such that t Zs (φ2 ) − Zs (φ)2 ds.

N (φ)t = 0

Superprocesses (b) Show that


∞

207

Z˜s (|Aφ|)ds < ∞ a.s. and then use this to prove that Nt (φ) con-

0

a.s.

verges a.s. as t → ∞ for all φ in D(A). Conclude that Zt (·) → Z∞ (·) in MF (E∞ ) for some Z∞ ∈ MF (E∞ ). (c) Prove that Z∞ = δF a.s. for some random point F in E∞ and hence conclude a.s. that Xt (·)/Xt (1) → δF (·) as t ↑ ζ. Hint. Prove lim Zt (φ2 )−Zt (φ)2 = 0 for all φ in D(A), by using the a.s. convergence t→∞

of Nt (φ). (d) Use Exercise II.5.4 to see that   PX0 F ∈ A | Xs (1), s ≥ 0 (ω) = P X0 (Yζ(ω) ∈ A)/X0 (1) a.s. for all A ∈ B(E∞ ), and in particular F ∈ E a.s. 3. Hausdorff Measure Properties of the Supports. Definition. If h : R+ → R is strictly increasing and continuous near 0 and h(0) = 0 (write h ∈ H), the Hausdorff h-measure of A ⊂ Rd is h − m(A) = lim inf δ↓0

∞ ∞      h diam(Bi ) : A ⊂ Bi , Bi balls with diam(Bi ) ≤ δ i=0

1

The Hausdorff dimension of A is dim(A) = inf{α : xα − m(A) < ∞}(≤ d). The first result gives a global (in time) upper bound on S(Xt ) for d ≥ 3 which allows one to quickly understand the 2-dimensional nature of S(Xt ). Until otherwise indicated (Xt , t ≥ 0) is a SBM(γ) starting at X0 ∈ MF (Rd ) under PX0 , Pt is the Brownian semigroup and A is its (weak) generator. Proposition III.3.1. Let ψ(r) = r 2 (log+ 1/r)−1 and d ≥ 3. Then   ψ − m S(Xt ) < ∞ ∀ t > 0 a.s. and, in particular, dimS(Xt ) ≤ 2 ∀t > 0 a.s. Proof. By the Historical Cluster Representation (Theorem III.1.1)   j,n S Hj/2n ({y : y((j − 1)2−n ) ∈ ·) = {xj,n 1 . . . , xM (j,n) }, H where conditional on F(j−1)2 −n , M (j, n) has a Poisson distribution with mean n+1 −1 γ H(j−1)2−n (1) (j, n ∈ N). The Historical Modulus of Continuity (Theorem 2 III.1.3) implies that for a.a. ω if 2−n < δ(ω, 3) and t ∈ [j/2n , (j + 1)/2n ], then −n

(III.3.1)

(Corollary III.1.5) S(Xt ) ⊂ S(Xj2−n )3h(2 )  " #3h(2−n ) = Πj2−n S(Hj2−n ) (Corollary III.1.4(b)) 

M (j,n)

⊂

i=1

  −n B xj,n ) . i , 6h(2

208

Superprocesses

(The historical modulus is used in the last line.) A simple tail estimate for the Poisson distribution gives   n n2 .  / M (j, n) > 2n+2 γ −1 H(j−1)2−n (1) + 1  P j=1 n

≤

n2 

 . / . / E exp −2n+2 γ −1 (H(j−1)2−n (1) + 1) exp H(j−1)2−n (1)2n+1 γ −1 (e − 1)

j=1

  ≤ n2n exp −2n+2 γ −1 . By the Borel-Cantelli Lemma w.p.1 for large enough n (III.3.1) holds, and for all j2−n ≤ n, 

M (j,n)

        ψ 12h(2−n ) ≤ sup Ht (1) + 1 2n+2 γ −1 ψ 12h(2−n ) ≤ c sup Ht (1) + 1 . t

i=1

t

  This implies ψ − m S(Xt ) < ∞ ∀t > 0 a.s. because Xt = 0 for t large enough. Remark III.3.2. By taking unions over j ≤ 2n K in (III.3.1) we see from the above argument that for K ∈ N and 2−n < δ(ω, 3)  t∈[2−n ,K]

S(Xt ) ⊂

n (j,n) K2  M

j=1

  −n ) B xj,n i , 6h(2

i=1

  and for n large enough this is a union of at most K0 22n+2 γ −1 sup Ht (1) + 1 balls t

of radius 6h(2−n ). As Xt = 0 for t > K(ω), this shows that f − m(R([δ, ∞))) < ∞ ∀δ > 0 where f (r) = r 4 (log 1/r)−2 , and so dim R ≤ 4. A refinement of this result which in particular shows that R is Lebesgue null in the critical 4-dimensional case is contained in Exercise III.5.1 below. The exact results are described below in Theorem III.3.9. In order to obtain more precise Hausdorff measure functions one must construct efficient coverings by balls of variable radius (unlike those in Proposition III.3.1). Intuitively speaking, a covering is efficient if the balls contain a maximal amount of Xt -mass for a ball of its radius. This suggests that the lim sup behaviour of Xt B(x, r) as r ↓ 0 is critical. The following result of Rogers and Taylor (1961) (see Perkins (1988) for this slight refinement) plays a central role in the proof of the exact results described below (see Theorems III.3.8 and III.3.9). Proposition III.3.3. There is a c(d) > 0 such that for any h ∈ H, K > 0 and ν ∈ MF (Rd ), (a) ν(A) ≤ Kh − m(A) whenever A is a Borel subset of   E1 (ν, h, K) = {x ∈ Rd : lim ν B(x, r) /h(r) ≤ K}. r↓0

Superprocesses

209

(b) ν(A) ≥ c(d)Kh − m(A) whenever A is a Borel subset of   E2 (ν, h, K) = {x ∈ Rd : lim ν B(x, r) /h(r) ≥ K}. r↓0

We can use Proposition III.3.3 (a) to get a lower bound on S(Xt ) which complements the upper bound in Proposition III.3.1. Notation. If d ≥ 3, let hd (r) = r 2 log+

1 r

and define h2 (r) = r 2 (log+ 1r )2 .

Theorem III.3.4. If d ≥ 2 there is a c(d) > 0 such that for all X0 ∈ MF (Rd ), PX0 -a.s.   ∀δ > 0 there is an r0 (δ, ω) > 0 so that sup Xt B(x, r) ≤ γc(d)hd (r) ∀r ∈ (0, r0 ). x,t≥δ

This result is very useful when handling singular integrals with respect to Xt (e.g., see the derivation of the Tanaka formula in Barlow-Evans-Perkins (1991)). Before proving this, here is the lower bound on S(Xt ) promised above.   Corollary III.3.5. If d ≥ 2, Xt (A) ≤ γc(d)hd − m A ∩ S(Xt ) ∀ Borel set A and t > 0 PX0 -a.s. ∀X0 ∈ MF (Rd ). In addition, if ζ = inf{t : Xt = 0}, then dim S(Xt ) = 2 for 0 < t < ζ PX0 -a.s. ∀X0 ∈ MF (Rd ). Proof. By the previous result we may apply Proposition III.3.3 (a) to the sets A ∩ S(Xt ) for all Borel A, t > 0 to get the first inequality. This with Proposition III.3.1 together imply that for d ≥ 2, dim S(Xt ) = 2 ∀0 < t < ζ a.s.
t Notation. If f ≥ 0 is Borel measurable, let G(f, t) = sup Ps f (x)ds. x

0

Lemma III.3.6. If f is a non-negative Borel function such that G(f, t)γ/2 < 1, then     γ PX0 exp (Xt (f )) ≤ exp X0 (Pt f )(1 − G(f, t))−1 < ∞. 2  γ −1 Proof. Let k(s) = 1− 2 G(f, t−s) and φ(s, x) = Pt−s f (x)k(s) (t > 0, s ∈ (0, t]).  t−ε . To see this note that Pε f ∈ D(A) so that If ε > 0, we claim φ |[0,t−ε]×Rd ∈ D(A) ∂ ∂ d ∂s Pt−s f = ∂s Pt−ε−s (Pε f ) = −Pt−ε−s (APε f ) is continuous on R and is bounded ∂φ d on [0, t − ε] × R . The same is true of ∂s (s, x), and clearly Pt−s f = Pt−s−ε (Pε f ) ∈ D(A) implies φ(s, ·) ∈ D(A) and Aφs = k(s)Pt−s−ε (APε f ) is bounded on [0, t − ε] × Rd . By Proposition II.5.7, for s < t Zs ≡ Xs (Pt−s f )ks s = X0 (Pt f )k0 +

s Xr (Pt−r f )k˙ r dr.

Pt−r f (x)kr M (dr, dx) + 0

0

210

Superprocesses

By Itˆ o’s lemma there is a continuous local martingale Ns with N0 = 0 so that for s < t,   eZs = exp X0 (Pt f )k0 + Nt +

s

 γ eZr Xr (Pt−r f )k˙ r + (Pt−r f )2 kr2 dr. 2

0

Our choice of k shows the quantity in square brackets is less than or equal to 0. This shows that eZs is a non-negative local supermartingale, and therefore a supermartingale by Fatou’s Lemma. Fatou’s lemma also implies E(eZt ) ≤ lim inf E(eZs ) ≤ eZ0 , s↑t

which gives the result. Remark III.3.7. The above proof shows the Lemma holds for f ∈ D(A)+ for any BSMP Y satisfying (II.2.2), or for f ∈ bB(Rd )+ and any BSMP Y satisfying (II.2.2) and Pt : bB(Rd )+ → D(A) for t > 0. (We still assume g = 0, γ constant.) In particular Lemma III.3.6 is valid for the α-stable-DW-superprocess. Schied (1996) contains more on exponential inequalities as well as the idea underlying the above argument. Proof of Theorem III.3.4. Recall h(r) = (r log 1/r)1/2 . We first control   −n Xt B(x, h(2 )) using balls in    √  Bn = B x0 , ( d + 4)h(2−n ) : x0 ∈ 2−n/2 Zd ∩ [−n, n]d . Here εZd = {εn : n ∈ Zd }. Assume 2−n < δ(ω, 3), where δ(ω, 3) is as in the Historical Modulus of Continuity. n]d , choose x0 ∈ 2−n/2 Zd ∩ [−n, n]d √ −n/2 If√x ∈ [−n, −n such that |x0 − x| < d 2 ≤ d h(2 ) if n ≥ 3. Assume j ∈ Z+ and j2−n ≤ t ≤ (j + 1)2−n . For Ht -a.a. y if y(t) ∈ B x, h(2−n ) , then |y(j2−n ) − x0 | ≤ |y(j2−n ) − y(t)| + |y(t) − x| + |x − x0 | √ ≤ (4 + d)h(2−n ). √   If B = B x0 , (4 + d)h(2−n ) ∈ Bn , this shows that   /   . Xt B(x, h(2−n )) = Ht y(t) ∈ B(x, h(2−n )) ≤ Ht y(j2−n ) ∈ B ≡ Mj2−n ,B (t − j2−n ). Lemma III.1.2 and the Markov property show that Mj2−n ,B (t) is a martingale and so Mj2−n (t) = supB∈Bn Mj2−n ,B (t) is a non-negative submartingale. The above bound implies   sup sup Xt B(x, h(2−n )) ≤ sup Mj2−n (t) ∀ j ∈ Z+ −n −n x∈[−n,n]d t≤2−n (III.3.2) j2 ≤t≤(j+1)2 whenever 2−n ≤ δ(ω, 3).

Superprocesses

211

We now use the exponential bound in Lemma III.3.6 to bound the right side of the above. Let β = β(d) = 1 + 1(d = 2) and set εn = c1 2−n n1+β(d) where c1 will be chosen (large enough) below. An easy calculation shows that  +   √     (III.3.3) G B(x, r), t ≡ G 1B(x,r) , t ≤ c2 (d)r 2 1 + 1(d = 2) log( t/r) from which it follows that (III.3.4)

sup B∈Bn ,t≤n

G(B, t) ≤ c3 (d)2−n nβ(d) .

−1 n−1 −β(d) 2 n , then the weak maximal inequality for subIf δ > 0 and λn = c−1 3 γ martingales implies (III.3.5)  

PX0

sup

δ2n ≤j
≤ n2n

sup Mj2−n (t) ≥ εn

t≤2−n

  −n e−λn εn PX0 eλn Mj2−n (2 )

sup

δ2n ≤j
≤ n2n |Bn |

sup

δ2n ≤j
   e−λn εn PX0 exp Xj/2n (B)2λn (2 − λn γ2−n )−1 ,

where we have used Lemma III.1.2, the Markov property and Lemma III.3.6 with f ≡ λn . The latter requires λn 2−n γ/2 < 1 which is certainly true for n ≥ n0 . In fact for n ≥ n0 (III.3.5) is no more than    sup sup e−λn εn PX0 exp 2λn Xj/2n (B) n2n |Bn | (III.3.6)

δ2n ≤j
−λn εn

≤ n2 |Bn |e n

B∈Bn

sup

δ2n ≤j
  sup exp 4λn P X0 (Bj/2n ∈ B)

B∈Bn

by another application of Lemma III.3.6, this time with f = 2λn 1B . We also use (III.3.4) here to see that 2λn G(B, j/2−n ) ≤ 2λn c3 2−n nβ(d) = γ −1 , so that Lemma III.3.6 may be used and the upper bound given there simplifies to the expression in (III.3.6). An elementary bound shows that the right side of (III.3.6) is at most   cn1+d 2n+nd/2 e−λn εn exp 4λn c(δ, X0 (1))h(2−n )d     −1 −1 2 n c δ, X0 (1) , ≤ cn1+d 2n(1+d/2) exp −c1 c−1 3 γ   which is summable if c1 = γ2c3 1 + d2 ≡ γc (d). Borel-Cantelli and (III.3.2) imply that a.s. for large enough n,     sup Xt B(x, h(2−n )) < γc 2−n n1+β(d) ≤ γc hd h(2−n ) . sup δ≤t≤n x∈[−n,n]d

An elementary interpolation completes the proof. In view of Proposition III.3.1 and Corollary III.3.5 it is natural to ask if there is an exact Hausdorff measure function associated with S(Xt ).

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Notation. * 2 r log+ log+ 1/r hd (r) = r 2 (log+ 1/r)(log+ log+ log+ 1/r)

d≥3 d = 2.



log+ x = (log x) ∨ ee



Theorem III.3.8. Assume X is a SBM(γ) starting at µ under Pµ . (a) [d ≥ 2, t fixed] There is a universal constant c(d) > 0 such that ∀µ ∈ MF (Rd ), t>0   Xt (A) = γc(d)hd − m A ∩ S(Xt ) ∀A ∈ B(Rd ) Pµ -a.s. (b) [d ≥ 2, t variable] There are universal constants 0 < c(d) ≤ C(d) < ∞ such that for any µ ∈ MF (Rd )     (i) If d ≥ 3, γc(d)hd − m A ∩ S(Xt ) ≤ Xt (A) ≤ γC(d)hd − m A ∩ S(Xt ) ∀A ∈ B(Rd ) ∀t > 0 Pµ -a.s.     (ii) If d = 2, γc(2)h3 − m A ∩ S(Xt ) ≤ Xt (A) ≤ γC(2)h2 − m A ∩ S(Xt ) ∀A ∈ B(Rd ) ∀t > 0 Pµ -a.s. (c) [d = 1] There is a jointly continuous process {u(t, x) : t > 0, x ∈ R} such that Xt (dx) = u(t, x)dx ∀t > 0 Pµ -a.s. Remarks. (1) (b) shows that if d ≥ 2 then w.p.1 for any t > 0 S(Xt ) is a singular set of Hausdorff dimension 2 whenever it is non-empty. This fact has already been proved for d ≥ 3 (Corollary III.3.5). (2) (a) and (b) state that Xt distributes its mass over S(Xt ) in a deterministic manner. This extreme regularity of the local structure of S(Xt ) is due to the fact that for d ≥ 2 the local density of mass at x is due entirely to close cousins of “the particle at x” and so will exhibit strong independence in x. The strong recurrence for d = 1 means this will fail in R1 and a non-trivial density, u, exists. (3) We conjecture that one may take c(d) = C(d) in (b)(i) for d ≥ 3. The situation in the plane is much less clear. (4) Curiously enough the exact Hausdorff measure functions for S(Xt ) are exactly the same as those for the range of a Brownian path (see Ciesielski-Taylor (1962), Taylor (1964)) although these random sets certainly look quite different. The two sets behave differently with respect to packing measure: h(s) = s2 (log log 1/s)−1 is an exact packing measure function for the range of a Brownian path for d ≥ 3 while h(s) = s2 (log 1/s)−1/2 is critical for S(Xt ) for d ≥ 3 (see Le Gall-Perkins-Taylor (1995)). (5) (b) is proved in Perkins (1988, 1989). In Dawson-Perkins (1991), (a) was then proved for d ≥ 3 by means of a 0-1 law that the Radon-Nikodym  which showed  derivative of Xt with respect to hd − m · ∩S(Xt ) is a.s. constant. The more delicate 2-dimensional case in (a) was established in Le Gall-Perkins (1995) using the Brownian snake. This approach has proved to be a very powerful tool in the study of path properties of (and other problems associated with) DW-superprocesses. The d = 1 result was proved independently by Reimers (1989) and Konno-Shiga (1988) and will be analyzed in detail in Section III.4 below. The existence of a density at a fixed time was proved by Roelly-Coppoletta (1986). The first Hausdorff measure result, dimS(Xt ) ≤ 2, was established by Dawson and Hochberg (1979).

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213

Here are the exact results for the range of super-Brownian motion, promised after Proposition III.3.1. Theorem III.3.9. (a) d ≥ 4. Let * r 4 log+ log+ 1/r ψd (r) = r 4 (log+ 1/r)(log+ log+ log+ 1/r)

d>4 . d=4

There is a c(d) > 0 such that for all X0 ∈ MF (Rd ), t Xs (A)ds = γc(d)ψd − m (A ∩ R((0, t])) ∀A ∈ B(Rd ) ∀t ≥ 0 PX0 − a.s. 0

(b) d ≤ 3. Assume X0 has a bounded density if d = 2, 3. Then there is a jointly continuous density {v(t, x) : t ≥ 0, x ∈ Rd } such that

t

v(t, x)dx ∀A ∈ B(Rd ) ∀t ≥ 0

Xs (A)ds = 0

PX0 − a.s.

A

Discussion. (a) for d > 4 is essentially proved in Dawson-Iscoe-Perkins (1989). There upper and lower bounds with differing constants were given. Le Gall (1999) showed that by a 0 − 1 law the above constants were equal and used his snake to derive the critical 4-dimensional result. (b) is proved in Sugitani (1989). Consider now the analogue of Theorem III.3.8 when X is an α-stable-DWsuperprocess (g ≡ 0, γ(·) ≡ γ > 0). That is Y is the symmetric α-stable process in Rd considered in Example II.2.4 (b). Let hd,α (r) = r α log+ log+ 1/r α

+

2

hd,α (r) = r (log 1/r) * α r ψd,α (r) = r α log+ 1/r

if d > α

if d = α if d > α . if d = α

In Perkins (1988) it is shown that for d > α there are real constants 0 < c1 ≤ c2 , depending only on (d, α) so that   Xt B(x, r) (III.3.7) γc1 ≤ lim ≤ γc2 Xt -a.a. x ∀t > 0 PX0 -a.s. r↓0 hd,α (r) Let Λt (ω) be the set of x for which the above inequalities hold. Then Λt (ω) is a Borel set supporting Xt (ω) for all t > 0 a.s. and Proposition III.3.3 shows that (for d > α) (III.3.8)

γc(d)c1 hd,α − m(A ∩ Λt ) ≤ Xt (A) ≤γc2 hd,α − m(A ∩ Λt ) ∀A ∈ B(Rd ), t > 0 PX0 -a.s.

For t fixed a 0 − 1 law then shows the lim in (III.3.7) is γc3 for Xt -a.a. x, say for x ∈ Λt , a.s. and Xt (·) = c4 γhd,α − m(· ∩ Λt ) a.s. for some c3 , c4 > 0 (see Theorem

214

Superprocesses

5.5 of Dawson-Perkins (1991)). Analogous results are also shown for d = α (=1 or 2) with ψd,α − m in the lower bound and hd,α − m in the upper bound. Such results are clearly false for α < 2 if Λt is replaced by S(Xt )(= φ or Rd a.s. by Example III.2.3). (III.3.7) suggests we define the Campbell measure, Qt ∈ MF (MF (Rd ) × Rd ), associated with Xt by Qt (A × B) = 1A (Xt )Xt (B)dPX0 . If (X, Z) ∈ MF (Rd ) × Rd are coordinate variables on MF (Rd ) × Rd ,then under Qt , Z is chosen at random according to X. The regular conditional probabilities Qt (X ∈ · | Z = x) are the Palm measures of Xt and describe Xt from the perspective of a typical point x in the support of Xt (see Dawson-Perkins (1991), Chapter 4, and Dawson (1992), Chapter 6, for more on Palm measures of superprocesses).  The first  step in deriving (III.3.7) is to find the asymptotic mean size of Xt B(x, r) when x is chosen according to Xt . (a) of the following Exercise is highly recommended. Exercise III.3.1. Let X be the α-stable-DW-superprocess and Qt (dX, dZ) be the Campbell measure defined above. If Y is the α-symmetric stable process in Rd , Yt has a smooth symmetric density pt (y) = pt (|x|) such that p1 (·) is decreasing on [0, ∞) and p1 (r) ≤ c(1 + r)−(d+α) . (a) If d ≥ α, show there is a constant kd,α > 0 such that  
  
 X B(Z, r) Qt (dX, dZ) E Xt B(x, r) dXt (x) lim ≡ lim r↓0 r↓0 ψd,α (r) ψd,α (r) (III.3.9) = kd,α γX0 (1) ∀t > 0.   Also show there is a c = c d, α, γ, X0(1) such that for any δ > 0 there is an r0 (δ) > 0 satisfying    (III.3.10) E Xt B(x, r) dXt (x) ≤ cψd,α (r) ∀t ∈ [δ, δ −1 ], r ≤ r0 (δ). (b) Show the above results remain valid for d < α if ψd,α (r) = r, kd,α may depend on (t, γ, X0), and c may also depend on δ.
(c) [Palm measure version] If X0 Pt (x) = pt (y − x)X0 (dy), show that Qt (X (B(Z, r)) | Z = x0 ) γE X0 = P X0 (Yt ∈ B(x0 , r)) +


t

   P Ys |Yt−s − x0 | < r pt−s (x0 − Ys )ds

0

X0 Pt (x0 )

.

*(d) Show there is a constant kd,α (which may also depend on (γ, t, x0, X0 ) if d < α) such that for any t > 0, x0 ∈ Rd ,     Qt X B(Z, r) | Z = x0 = γkd,α . (III.3.11) lim r↓0 ψd,α (r)

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215

Recall that ψd,α (r) = r if d < α.

  The above Exercise shows that if d > α, the mean of Xt B(x, r) when x is chosen according to Xt behaves like kd,α ψd,α (r) as r ↓ 0. This explains the rα part of hd,α in (III.3.7). The log log 1/r then comes from the exponential tail of Xt B(x, r) and the fact that it suffices to consider r ↓ 0 through a geometric sequence (each exponentiation produces a log). It is an easy Borel-Cantelli Exercise to use the above mean results to obtain weaker versions of the upper bounds in (III.3.7) and (III.3.8). Exercise III.3.2. (a) Use (III.3.10) to show that if d ≥ α, then for any ε > 0, t > 0   Xt B(x, r) = 0 Xt -a.a. x PX0 -a.s. lim r↓0 ψd,α (r)(log+ 1/r)1+ε (b) If φεd,α (r) = ψd,α (r)(log+ 1/r)1+ε , show that ∀t > 0, φεd,α − m(A) = 0 implies

Xt (A) = 0

∀A ∈ B(Rd ), ε > 0 PX0 -a.s.

4. One-dimensional Super-Brownian motion and Stochastic PDE’s In this section we study super-Brownian motion in one dimension with constant branching rate γ > 0. In particular, we will prove Theorem III.3.8(c) and establish a one-to-one correspondence between the density of super-Brownian motion in one dimension and the solution of a parabolic stochastic pde driven by a white noise on R+ × R. Let Ft be a right continuous filtration on (Ω, F , P ) and let P = P(F· ) denote the σ-field of Ft -predictable sets in R+ × Ω. Let |A| denote the Lebesgue measure of a Borel set in Rd and let BF (Rd ) = {A ∈ B(Rd ) : |A| < ∞}. Definition. An (Ft )-white noise, W , on R+ × Rd is a random process {Wt (A) : t > 0, A ∈ BF (Rd )} such that (i) Wt (A ∪ B) = Wt (A) + Wt (B) a.s. for all disjoint A, B ∈ BF (Rd ) and t > 0. (ii) ∀A ∈ BF (Rd ), t → Wt (A) is an (Ft )-Brownian motion starting at 0 with diffusion parameter |A|. See Chapters 1 and 2 of Walsh (1986) for more information on white noises. Note that if W is above and A and B are disjoint sets in BF , then 2Wt (A)Wt (B) = Wt (A ∪ B)2 − Wt (A)2 − Wt (B)2 is an (Ft )-martingale. It follows that W is an orthogonal martingale measure in the sense of Chapter 2 of Walsh (1986) and Proposition 2.10 of Walsh (1986) shows that the above definition is equivalent to that given in Walsh (1986). As in Section II.5 (see Chapter 2 of Walsh (1986)) one can define a stochastic integral
t
Wt (ψ) = 0 ψ(s, ω, x)dW (s, x) for

216

Superprocesses

ψ ∈ L2loc (W ) ≡{ψ : R+ × Ω × Rd → R : ψ is P × B(Rd ) − measurable, t ψ(s, ω, x)2dx ds < ∞ ∀t > 0 a.s. }. 0

The map ψ → W (ψ) is a linear map from L2loc to the space of continuous Ft -local martingales and Wt (ψ) has predictable square function t W (ψ)t = ψ(s, ω, x)2dx ds for all t > 0 a.s. 0

Notation. If f, g are measurable functions on Rd , let f, g = integral exists.




f (x)g(x) dx if the

¯ = (Ω, F , Ft, P ), Definition. Let W be an (Ft )-white noise on R+ ×Rd defined on Ω d let m ∈ MF (R ) and let f : R+ → R. We say that an adapted continuous process u : (0, ∞) × Ω → CK (Rd )+ is a solution of ∂u ∆u √ ˙ + f (u), = + γuW ∂t 2

(SP DE)fm

u0+ (x)dx = m

¯ iff for every φ ∈ C 2 (Rd ), on Ω b t t

∆φ  ds + us , φ(x) γu(s, x)dW (s, x) ut , φ = m(φ) + 2 0 0 t f (u(s, x))φ(x)dx ds for all t > 0 a.s. + 0

Remark III.4.1. Use the fact that Cb2 (Rd ) contains a countable convergence determining class (such as {sin(u · x), cos(u · x) : u ∈ Qd }) to see that (SPDE)fm implies that limt→0+ ut (x)dx = m a.s. in MF (Rd ). We have been able to choose a rather restrictive state space for ut because the Compact Support Property for SBM (Corollary III.1.4) will produce solutions with compact support. This property will persist for stochastic pde’s in which the square root in the white noise integrand is replaced by any positive power less than 1, but fails if this power equals 1 (see Mueller and Perkins (1992) and Krylov (1997)). We write (SPDE)m for (SPDE)0m . The one-dimensional Brownian density and semigroup are denoted by pt and Pt , respectively. We set pt (x) = 0 if t ≤ 0. Theorem III.4.2. (a) Let X be an (Ft )-SBM(γ) in one spatial dimension, starting ¯  = (Ω , F  , F  , P  ). There is an adapted continuous at X0 ∈ MF (R) and defined on Ω t CK (R)-valued process {ut : t > 0} such that Xt (dx) = ut (x) dx for all t > 0 P  -a.s. Moreover for all t > 0 and x ∈ R, t (III.4.1) ut (x) = Pt X0 (x) + pt−s (y − x)dM (s, y) P  − a.s., and 0 

v 2  < ∞. E  sup pt−s (y − x)dM (s, y) v≤t

0

Superprocesses

217

¯  = (Ω , F  , F  , P  ) so (b) Let X and u be as above. There is a filtered space Ω t that ¯ = (Ω, F , Ft, P ) ≡ (Ω × Ω , F  × F  , (F· × F· )t+ , P  × P  ) Ω ¯ where carries an Ft -white noise, W , on R+ × R and u ◦ Π satisfies (SPDE)X0 on Ω,  Π : Ω → Ω is the projection map. ¯ = (Ω, F , Ft, P ). Then (c) Assume u satisfies (SPDE)m (d-dimensional) on some Ω * u(t, x)dx if t > 0 Xt (dx) = m(dx) if t = 0 ¯ starting at m. defines an (Ft )-SBM(γ) on Ω A proof is given below. Clearly Theorem III.3.8(c) is immediate from (a). Corollary III.4.3. (a) If d = 1, then for any m ∈ MF (R) there is a solution to (SPDE)m and the law of u on C((0, ∞), CK (R)) is unique. (b) If d ≥ 2 and m = 0, there is no solution to (SPDE)m . Proof. (a) The existence is included in Theorem III.4.2(b). The Borel subsets of CK (R) are generated by the coordinate maps. To prove uniqueness in law it therefore suffices to show that if u satisfies (SPDE)m , then (III.4.2) P ((uti (xi ))i≤n ∈ ·) is unique on B(Rn ) for any 0 < ti , xi ∈ R, n ∈ N. If X is the SBM(γ) in Theorem III.4.2(c), then uti (xi ) = limε→0 Xti (pε (· − xi )), and the uniqueness in law of the super-Brownian motion X clearly implies (III.4.2). (b) If u is a solution to (SPDE)m for d ≥ 2, then by Theorem III.4.2(c), Xt (dx) = u(t, x)dx for t > 0 and X0 = m defines a super-Brownian motion starting at m which is absolutely continuous for t > 0. This contradicts the a.s. singularity of super-Brownian motion in dimensions greater than 1 (Theorem III.3.8(a), but note that Proposition III.3.1 suffices if d ≥ 3). The uniqueness in law for the above stochastic pde does not follow from the standard theory since the square root function is not Lipschitz continuous. For √ Lipschitz continuous functions of the solution (as opposed to γu) solutions to (SPDE) are unique in law when the initial condition has a nice density (see Chapter 3 of Walsh (1986)). This needs naturally to Open Problem. Does pathwise uniqueness hold in (SPDE)m ? That is, if u, v are solutions of (SPDE)m with the same white noise W , is it true that u = v a.s.? Note that the finite-dimensional version of this problem is true by Yamada-Watanabe (1971). Recently Mytnik (1998) proved uniqueness in law for solutions of (SPDE)m when the square root in front of the white noise is replaced by a power between 1/2 and 1. His argument may be viewed as an extension of the exponential duality used to prove uniqueness for superprocesses but the dual process is now random. It does apply for slightly more general functions than powers but, as is often the case with duality arguments, the restriction on the functions is severe and artificial. This is one reason for the interest in the above problem as a pathwise uniqueness argument would likely be quite robust.

218

Superprocesses

To prove Theorem III.4.2 we use the following version of Kolmogorov’s continuity criterion for two-parameter processes. Proposition III.4.4. Let I : (t0 , ∞) × R → R be a process on (Ω, F , P ) such that for some p > 1, a, b > 2, for any T > t0 , there is a c = c(T ) so that E(|I(t , x ) − I(t, x)|p) ≤ c(T )[|t − t|a + |x − x|b ] ∀t, t ∈ (t0 , T ], x, x ∈ [−T, T ]. Then I has a continuous version. Proof. See Corollary 1.2 of Walsh (1986) where a modulus of continuity of the continuous version is also given. Lemma III.4.5. (a) If 0 ≤ δ ≤ p, then |pt+ε (z) − pt (z)|p ≤ (εt−3/2 )δ [pt+ε (z)p−δ + pt (z)p−δ ]

∀z ∈ R, t > 0, ε ≥ 0.

(b) If 0 < δ < 1/2, there is a c(δ) > 0 so that for any 0 ≤ t ≤ t ≤ T and x, x ∈ R,

t



(pt −s (y − x ) − pt−s (y − x))2 dy ds ≤ |x − x| + c(δ)T 1/2−δ |t − t|δ .

0

Proof. (a) By the mean value theorem there is a u ∈ [t, t + ε] such that       ∂pu  pu (z)  z 2  (z) = ε − 1 |pt+ε (z) − pt (z)| = ε    ∂u 2u u ≤ εu−3/2 ≤ εt−3/2 , where a calculus argument has been used to see that ( )  2   z √ 1 upu (z)/2  − 1 ≤ (2π)−1/2 sup xe−x ∨ ≤ 1. u 2 x>1/2 Therefore for 0 ≤ δ ≤ p, |pt+ε (z) − pt (z)|p ≤ (εt−3/2 )δ |pt+ε (z) − pt (z)|p−δ " # ≤ (εt−3/2 )δ pt+ε (z)p−δ + pt (z)p−δ . (b) Note that if 0 ≤ t ≤ t ≤ T , then

t



(pt −s (y − x ) − pt−s (y − x))2 dy ds

0

=

t



pt −s (y − x )2 dy ds +

t

t

(pt −s (y − x ) − pt−s (y − x))2 dy ds

0

(III.4.3) ≡ I1 + I2 . By Chapman-Kolmogorov, (III.4.4) I1 = t

t

p2(t −s) (0) ds = π −1/2 (t − t)1/2 .

Superprocesses

219

If we expand the integrand in I2 , let ∆ = x − x, and use Chapman-Kolmogorov, we get t p2(t −s) (0) + p2(t−s) (0) − 2pt −s+t−s (∆) ds I2 = 0 t t p2(t −s) (0) − pt −s+t−s (∆) ds + p2(t−s) (0) − pt −s+t−s (∆) ds = 0

0

≡ I3 + I4 . Consider I3 and use (a) with p = 1 and 0 < δ < 1/2 to see that t (p2(t −s) (0) − p2(t −s) (∆)) + (p2(t −s) (∆) − pt −s+t−s (∆)) ds I3 = 0

1 ≤ 2



2t

ps (0) − ps (∆) ds t " # + (t − t)δ (t − s)−δ3/2 p2(t −s) (∆)1−δ + pt +t−2s (∆)1−δ ds 0

0



2t

s−1/2 [1 − exp(−∆2 /(2s))] ds ≤ 2−3/2 π −1/2 0 t (t − s)−δ3/2−(1−δ)/2 (4π)−(1−δ)/2 2 ds + (t − t)δ 0 ∞ t −1 −3/2 −u  δ u (1 − e ) du + 1.1(t − t) s−1/2−δ ds ≤ (4π) |∆| ≤ (4π)−1 |∆|

∆2 /2t ∞ −3/2



u

0

(1 ∧ u) du + 1.1(1/2 − δ)−1 t1/2−δ (t − t)δ

0

≤ π −1 |∆| + c (δ)T 1/2−δ (t − t)δ . Use this, the analogous bound for I4 , and (III.4.4) in (III.4.3) to conclude that the left-hand side of (III.4.3) is bounded by π −1/2 |t − t|1/2 + |x − x| + c (δ)T 1/2−δ (t − t)δ ≤ |x − x| + c(δ)T 1/2−δ (t − t)δ . We let pxε (y) = pε (y − x). To use Proposition III.4.4 we will need the following bound on the moments of Xt . Lemma III.4.6. If X is as in Theorem III.4.2 (a), then E  (Xt (pxε )p ) ≤ p!γ p tp/2 exp(X0 (1)/γt) ∀t, ε > 0, x ∈ R, and p ∈ N. Proof. We apply Lemma III.3.6 with f = θpxε , where θ = γ −1 t−1/2 and ε, t > 0 and x ∈ R are fixed. Then γ γ t γθ t G(f, t) ≡ sup Ps f (x ) ds = ps+ε (0) ds ≤ 1/2, 2 2 0 x 2 0

220

Superprocesses

and so Lemma III.3.6 implies that E  (exp(θXt (pxε ))) ≤ exp(2θX0 (pxt+ε )) ≤ exp(X0 (1)/γt). This shows that for any p ∈ N, E  (Xt (pxε )p ) ≤ p!θ −p exp(X0 (1)/γt), as is required. Proof of Theorem III.4.2. (a) We adapt the argument of Konno and Shiga (1988). From (GFR) (see Exercise II.5.2) we see that for each fixed ε, t > 0 and x ∈ R, t pt+ε−s (y − x)dM (s, y) a.s. (III.4.5) Xt (pxε ) = X0 (pxt+ε ) + 0

Lemma III.4.5 (a) with δ = p = 1 implies that lim sup |X0 (pxt+ε ) − X0 (pxt )| ≤ 2εη −3/2 X0 (1) ∀η > 0.

(III.4.6)

ε↓0 x∈R,t≥η

To take L2 limits in the stochastic integral in (III.4.5) apply Lemma III.4.5(a) with p = 2 and 0 < δ < 1/2 to see that  t  (pt+ε−s (y − x) − pt−s (y − x))2 Xs (dy) ds E 0   t (pt+ε−s (y − x) − pt−s (y − x))2 ps (y − z) dy ds m(dz) = 0

t

(ε(t − s)−3/2 )δ [pt+ε−s (y − x)2−δ + pt−s (y − x)2−δ ] 0  × ps (y − z)dy ds m(dz) t ≤ εδ |t − s|−3δ/2 [(t + ε − s)(δ−1)/2 p(t+ε−s)/(2−δ) (y − x) 0  + (t − s)(δ−1)/2 p(t−s)/(2−δ) (y − x)]ps (y − z) dy ds m(dz) t  δ ≤ 2ε (t − s)−1/2−δ p(t+ε−s)/(2−δ)+s (x − z) 0   + p(t−s)/(2−δ)+s (x − z) ds m(dz)  t  −1/2 δ −1/2−δ −1/2 m(1)ε (t − s) s ds ≤ 4(2π) ≤

0

= m(1)c(δ)t−δ εδ . This implies that (III.4.7)  t  pt+ε−s (y − x) − pt−s (y − x)dM (s, y) lim sup E ε↓0 x∈R,t≥η

0

2

 = 0 ∀η > 0,

Superprocesses

221

and also shows that (III.4.8) u

pt−s (y − x)dM (s, y), u ≤ t is a continuous L2 -bounded martingale. 0

By (III.4.6) and (III.4.7) we may take L2 limits in (III.4.5) as ε ↓ 0 and also choose εn ↓ 0 so that for any t > 0 and x ∈ R, t pt−s (y − x)dM (s, y) a.s. and in L2 . lim Xt (pxεn ) = X0 (pxt ) + n→∞

0

Therefore if we define u(t, x) = lim inf n→∞ Xt (pxεn ) for all t > 0, x ∈ R, then t (III.4.9) u(t, x) = X0 (pxt )+ pt−s (y−x)dM (s, y) a.s. for all t > 0 and x ∈ R. 0

Also standard differentiation theory shows that for each t > 0 with probability 1, Xt (dx) = u(t, x)dx + Xts (dx), where Xts is a random measure such that Xts ⊥ dx. Now (III.4.8) and (III.4.9) imply that  E u(t, x) dx = X0 (pxt ) dx = X0 (1) = E  (Xt (1)). This shows that E  (Xts (1)) = 0 and so (III.4.10)

Xt (dx) = u(t, x)dx a.s. for all t > 0

Now fix t0 > 0. Apply (III.4.5) with t replaced by t0 and ε replaced by t−t0 > 0 to obtain t0 Xt0 (pxt−t0 ) = X0 (pxt ) + pt−s (y − x)dM (s, y) a.s. ∀t > t0 , x ∈ R. 0

This and (III.4.9) show that t u(t, x) = Xt0 (pxt−t0 ) + (III.4.11)

pt−s (y − x)dM (s, y) a.s. ∀t > t0 , x ∈ R t0

≡ Xt0 (pxt−t0 ) + I(t, x).

Proposition III.4.4 is now used to obtain a continuous version of I as follows. Let 0 < t0 < t ≤ t ≤ T , x, x ∈ R, and p > 1. Then Burkholder’s inequality and

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(III.4.10) show that (recall that pt (x) = 0 if t ≤ 0) E  (|I(t , x ) − I(t, x)|2p) p    t   2 ≤ cp E γ (pt −s (y − x ) − pt−s (y − x)) u(s, y) dy ds  ≤γ

t0 

t



p−1 

(pt −s (y − x ) − pt−s (y − x)) dy ds

p



t0 t

×



2

(pt −s (y − x ) − pt−s (y − x))2 E  (u(s, y)p) dy ds,

t0

by Jensen’s inequality. Lemma III.4.6 and Fatou’s Lemma show that (III.4.12)

E  (u(t, y)p) ≤ p!γ p tp/2 exp(X0 (1)/γt) ∀t > 0, x ∈ R, p ∈ N.

This, together with Lemma III.4.5(b) and the previous inequality, show that for any 0 < δ < 1/2,   E  |I(t , x ) − I(t, x)|2p ≤ cp γ p [|x − x | + c(δ)T 1/2−δ (t − t)δ ]p−1 (III.4.13)

× p!T p/2 exp(X0 (1)/γt0 )[|x − x | + c(δ)T 1/2−δ (t − t)δ ] ≤ c(p, γ, X0(1), t0 , T )[|x − x |p + |t − t|pδ ].

Proposition III.4.4 shows there is a continuous version of I on (t0 , ∞)×R. Dominated Convergence shows that (t, x) → Xt0 (pxt−t0 ) is also continuous on (t0 , ∞) × R a.s., and so (III.4.11) shows there is continuous version of u(t, x) on (t0 , ∞) × R for all t0 > 0 and hence on (0, ∞) × R. We also denote this continuous version by u. Clearly u(t, x) ≥ 0 for all (t, x) ∈ (0, ∞) × R a.s. Define a measure-valued process ˜ t (dx) = u(t, x)dx. Then (III.4.10) shows that by X (III.4.14)

˜ t = Xt a.s. for each t > 0. X

˜ t (φ) is continuous on (0, ∞) a.s. by Dominated ConverIf φ ∈ CK (R), then t → X gence and the continuity of u. Therefore the weak continuity of X and (III.4.14) imply ˜ t (φ) = Xt (φ) ∀t > 0 a.s., X and hence

˜ t (dx) ≡ u(t, x)dx ∀t > 0 a.s., Xt (dx) = X

where u is jointly continuous. Since ∪t≥η S(Xt ) is compact for all η > 0 a.s. by Corollary III.1.7, and u is uniformly continuous on compact sets, it follows easily that t → u(t, ·) is a continuous function from (0, ∞) to CK (R). (III.4.1) holds by (III.4.8) and (III.4.9), and so the proof of (a) is complete.

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(b) Let (Ω , F  , Ft , P  ) carry an (Ft )-white noise W  on R+ × R, and define W ¯ by on Ω t 1A (x)1(u(s, x) > 0)(γu(s, x))−1/2dM (s, x) Wt (A) = 0 t 1A (x)1(u(s, x) = 0)dW  (s, x), + 0

for t > 0 and A ∈ BF (R). The first stochastic integral is a continuous square inte
t
grable (Ft )-martingale with square function 0 1A (x)1(u(s, x) > 0) dx ds because the integrand is in the space L2 from Remark II.5.5(b). It is now easy to check that ¯ with square function |A|t and so is an Wt (A) is a continuous (Ft )-martingale on Ω (Ft )-Brownian motion with diffusion parameter |A|. Clearly if A, B are disjoint in BF (R), then Wt (A ∪ B) = Wt (A) + Wt (B) a.s. and so W is an (Ft )-white noise on R+ × R. We may extend the stochastic integrals with respect to M and W  to P(F· ) × B(R)-measurable integrands because M and W  are both orthogonal martingale measures with respect to Ft (we suppress the projection maps in our notation). It follows easily from the definition of W that if ψ ∈ L2loc (W ), then t

Wt (ψ) = 0

ψ(s, ω, x)1(u(s, x) > 0)(γu(s, x))−1/2dM (s, x) t ψ(s, ω, x)dW (s, x). + 0


t


then 0 φ(x)(γu(s, x))1/2dW (s, x) = Mt (φ). The martinTherefore if φ ∈ gale problem for Xt (dx) (= u(t, x)dx if t > 0) now shows that u satisfies (SPDE)m . Cb2 (R),

(c) The fact that t → u(t, ·) is a continuous CK (Rd )-valued process shows that Xt is a continuous MF (Rd )-valued process for t > 0. As was noted in Remark III.4.1, (SPDE)m implies that limt→0+ Xt = m a.s. and so X· is a.s. continuous on
t
[0, ∞). If φ ∈ Cb2 (Rd ) and Mt (φ) = 0 φ(x)(γu(s, x))1/2dW (s, x), then Mt (φ) is an Ft -local martingale satisfying t t 2 γφ(x) u(s, x) dx ds = Xs (γφ2 ) ds. M (φ)t = 0

0

Therefore (SPDE)m and Remark II.5.11 imply that Xt satisfies (LMP)δm and so X is an (Ft )-SBM(γ) starting at m. 5. Polar Sets Throughout  this section X is a SBM (γ) under PX0 , X0 = 0. Recall the range R([δ, ∞)). of X is R = δ>0

Definition. If A ⊂ Rd we say X(ω) charges A iff Xt (ω)(A) > 0 for some t > 0, and X(ω) hits A iff A ∩ R(ω) = φ. Theorem III.5.1. (a) If φ ∈ bB(Rd ), Xt (φ) is a.s. continuous on (0, ∞). (b) PX0 (X charges A) > 0 ⇔ A has positive Lebesgue measure. Proof. (a) See Reimers (1989b) or Perkins (1991), and the Remark below.

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(b) Since Xt (A) is a.s. continuous by (a), X charges A iff Xt (A) > 0 for some rational t > 0. The probability of the latter event is positive iff PX0 (Xt (A)) > 0 ∃t ∈ Q>0 ⇔ P X0 (Bt ∈ A) > 0 ∃t ∈ Q>0 ⇔ A has positive Lebesgue measure. Remark. In Theorem 4 of Perkins (1991) (a) is proved for a large class of Y – DW – superdiffusions whose semigroup satisfies a strong continuity condition. The simple idea of the proof is to first use the Garsia-Rodemich-Rumsey Lemma to obtain an explicit modulus of continuity for Xt (φ) − X0 (Pt φ) for φ ∈ Cb (E) and then to show bp

this modulus is preserved under → of φ. The strong continuity condition on Pt implies 1 Pt+r φ − Pt φr −1 dr < ∞ ∀t > 0, φ ∈ bE, 0

which is already stronger than the continuity of t → Pt φ(x) for each φ ∈ bE, x ∈ E. The latter is clearly a necessary condition for a.s. continuity of Xt (φ) on (0, ∞) (take means). Whether or not this, or even norm continuity of Pt φ ∀φ ∈ bE, is also sufficient for continuity of Xt (φ), t > 0 remains open. The notion of hitting a set will be probabilistically more subtle and more important analytically. Definition. A Borel set is polar for X iff PX0 (X hits A) = 0 for all X0 ∈ MF (Rd ) or equivalently iff PX0 (X hits A) = 0 for some non-zero X0 in MF (Rd ). The above equivalence is a consequence of the equivalence of PX0 and PX0 on  σ(Xr , r ≥ δ) for any non-zero finite measures X0 and X0 (see Example the field δ>0

III.2.3). We would like to find an analytic criterion for polarity. For ordinary Brownian motion Kakutani (1944) did this using Newtonian capacity. Definition. Let g be a decreasing non-negative continuous function on (0, ∞) with

g(|x − y|)µ(dx)µ(dy) is the g-energy of µ. If g(0+) > 0. If µ ∈ MF (Rd ), µg = A ∈ B(Rd ), let I(g)(A) = inf{µg : µ a probability, µ(A) = 1} and let the g-capacity at A be C(g)(A) = I(g)(A)−1 ∈ [0, ∞). Note that the g-capacity of A is positive iff A is large enough to support a probability of finite g-energy. Notation.

  r −β gβ (r) = 1 + (log 1/r)+  1

β>0 β=0 β<0

If φ ∈ H there is a close connection between sets of zero Hausdorff φ-measure and sets of zero φ−1 -capacity: (III.5.1) φ − m(A) < ∞ ⇒ C(φ−1 )(A) = 0. (III.5.2) C(gβ )(A) = 0 ⇒ xβ (log 1/x)−1−ε − m(A) = 0 ∀ε > 0, β > 0. (III.5.3) C(g0 )(A) = 0 ⇒ (log+ 1/x)−1 (log+ log+ 1/x)−1−ε − m(A) = 0 ∀ε > 0.

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Moreover these implications are essentially best possible. See Taylor (1961) for a discussion. In particular the capacitory dimension (defined in the obvious way) and Hausdorff dimension coincide. For d ≥ 2 the range of a Brownian motion {Bt : t ≥ 0} is two-dimensional and so should hit sets of dimension greater than d−2. Kakutani (1944) showed for d ≥ 2 and A ∈ B(Rd ), P (Bt ∈ A ∃t > 0) > 0 ⇔ C(gd−2 )(A) > 0. Recall from Theorem III.3.9 (see also Remark III.3.2) that R is a 4-dimensional set if d ≥ 4 and hence should hit sets of dimension greater than d − 4. Theorem III.5.2. Let A ∈ B(Rd ). A is polar for X iff C(gd−4 )(A) = 0. In particular, points are polar for X iff d ≥ 4. Remark III.5.3. The inner regularity of the Choquet capacities ¯ ∞)) = 0) (δ > 0) and A → C(gd−4 )(A) A → Pδx (A ∩ R([δ, (see III.29 of Dellacherie and Meyer (1978)) allows one to consider only A = K compact. The necessity of the zero capacity condition for polarity was proved in Perkins (1990) by a probabilistic inclusion-exclusion argument. The elegant proof given below is due to Le Gall (1999) (Section VI.2). The more delicate sufficiency was proved by Dynkin (1991). His argument proceeded in two steps: 1. V (x) = − log Pδx (R ∩ K = φ) is the maximal non-negative solution of ∆V = γV 2 on K c . 2. The only non-negative solution of ∆V = γV 2 on K c is V ≡ 0 iff C(gd−4 )(K) = 0. Step 1 uses a probabilistic representation of solutions to the non-linear boundary value problem ∆V = γV 2 in D, V |∂D = g, where D is a regular domain in Rd for the classical Dirichlet problem, and g is a bounded, continuous, non-negative function on ∂D. The representation is in terms of the exit measure XD of X on D. XD may be constructed as the weak limit of the of the sequence of measures obtained by stopping the branching particles in Section II.4 when they exit D and assigning mass 1/N to each exit location. Step 2 is the analytical characterization of the sets of removable singularities for ∆V = γV 2 due to Baras and Pierre (1984). A self-contained description of both steps may be found in Chapter VI of Le Gall (1999). A proof of a slightly weaker result is given below (Corollary III.5.10). The proof of the necessity of the zero capacity condition in Theorem III.5.2 will in fact show Theorem III.5.4. For any M > 0 and X0 ∈ MF (Rd )−{0}, there is a c(M, X0 ) > 0 such that PX0 (X hits A) ≥ c(M, X0 )C(gd−4 )(A) for any Borel subset A of B(0, M ). In particular points are not polar if d ≤ 3. Proof. We may assume A = K is a compact subset of B(0, M ) of positive gd−4 ν supported by capacity by the inner

regularity of C(gd−4 ). Choose a probability K so that E ≡ gd−4 (x − y)dν(x)dν(y) < ∞. Let f : Rd → R be a continuous, non-negative, radially symmetric function such that {f > 0} = B(0, 1) and

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f (y) dy = 1. Define fε (y) = ε−d f (ε−1 y) and φε (y) = fε ∗ ν(y) ≡ Note that




fε (y − z)ν(dz).

{φε > 0} ⊂ K ε ≡ {x : d(x, K) < ε}.

(III.5.4)

We will use the following elementary consequence of the Cauchy-Schwarz inequality. Lemma III.5.5. Assume Z ≥ 0 has mean µ and variance σ 2 < ∞. Then P (Z > 0) ≥ µ2 /(µ2 + σ 2 ). Proof. E(Z) = E(Z1(Z > 0)) ≤ E(Z 2 )1/2 P (Z > 0)1/2 . Rearrange to get a lower bound on P (Z > 0).
2
2 Apply the above lemma to Z = 1 Xs (φε )ds. If g1,2 (x) = 1 ps (x) ds, where ps is the Brownian transition density, then  2  2 PX0 Xs (φε ) ds = P X0 (φε (Bs )) ds 1 1  = g1,2 ∗ X0 (y)fε (y − z) dy ν(dz) → g1,2 ∗ X0 (z)ν(dz) as ε ↓ 0 > 0. The above shows that for ε < ε0 ,  1  2 inf g1,2 ∗ X0 (z) ≡ C1 (X0 , M ). Xs (φε ) ds ≥ (III.5.5) PX0 2 |z|≤M 1 By our second moment formula (see Exercise II.5.2(b))  2 2  2 Xs (φε ) ds = Cov (Xs (φε ), Xt(φε )) dsdt Var 1

1



1 2

= 2γ

dt

1

≤ 2γ



1

+

drX0 Pr (Ps−r φε Pt−r φε ) 0



1/2

0 2

s

ds



2

2

s

≡ 2γ[I1 + I2 ].

In I1 , s − r ≥ 1/2, t − r ≥ 1/2 and so



Pt−r φε ∨ Ps−r φε ≤ c

fε (y − z) dydν(z) = c.

This implies (the value of c may change) (III.5.7)

s

dtX0 Pr (Ps−r φε Pt−r φε )

ds r

1/2



2

dtX0 Pr (Ps−r φε Pt−r φε )

ds 1

dr



2

dr



(III.5.6)



t

I1 ≤ cX0 (1).



Superprocesses If G2 φε (y) =


2 0

Pt φε (y) dt, then

(III.5.8)

227



2

I2 ≤

X0 Pr (G2 φ2ε ) dr ≤ cX0 (1)

G2 φε (y)2 dy.

1/2


Lemma
III.5.6. (a) G2 φε (y) ≤ C gd−2 (y − z) dν(z). (b) gd−2 (z1 − y)gd−2 (z2 − y) dy ≤ Cgd−4 (z1 − z2 ).
2 Proof. (a) Since 0 pt (x)dt ≤ cgd−2 (x), we have gd−2 (y − x)fε (x − z) dxdν(z). G2 φε (y) ≤ c The superharmonicity of gd−2 implies that the spherical averages of gd−2 (y − x)fε
(x − z) over {x : |x − z)| = r} are at most gd−2 (y − z)fε (r). This and the fact that fε (y) dy = 1 allow us to conclude from the above that G2 φε (y) ≤ c gd−2 (y − z)dν(z). (b) Exercise. Kolmogorov.

One approach is to use gd−2 (x) ≤ c


1 0

pt (x) dt and Chapman-

Use (a) and (b) in (III.5.8) to see that gd−4 (z1 − z2 ) dν(z1 ) dν(z2 ) = cX0 (1)E. I2 ≤ cX0 (1) Now use the above with (III.5.5)–(III.5.8) in Lemma III.5.5 to see that  2  c(X0 , M ) C1 (X0 , M )2 ≥ , Xs (φε ) ds > 0 ≥ PX0 C1 (X0 , M )2 + cX0 (1)(1 + E) E 1 where we use E ≥ cM > 0 if K ⊂ B(0, M ) in the last line. Now minimize E to see that   2 PX0 Xs (φε ) ds > 0 ≥ c(X0 , M )C(gd−4 )(K). 1

This implies PX0 (R([1, 2]) ∩ K = φ) = lim PX0 (R([1, 2]) ∩ K ε = φ) ε↓0   2 ≥ lim PX0 Xs (φε ) ds > 0 , ε↓0

1

the last because S(φε ) ⊂ K . The above two inequalities complete the proof. ε

Upper bounds on hitting probabilities appear to require a greater analytic component. We now obtain precise asymptotics for hitting small balls using the Laplace functional equation (LE) from Section II.5. Recall from Theorem II.5.11 that if ∆/2

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denotes the generator of Brownian motion, f ∈ Cb (Rd )+ , and Vt is the unique solution of γ ∆ ∂V = Vt − Vt2 + f, V0 = 0, ∂t 2 2

(SE)0,f then (LE)



t

PX0 (exp{−

Xs (f ) ds}) = exp(−X0 (Vt )). 0

Recall also that f (ε) ∼ g(ε) as ε ↓ 0 means limε↓0 f (ε)/g(ε) = 1. Theorem III.5.7.(a) If d ≤ 3, then PX0 (X hits {x}) = 1 − exp



−

2(4 − d) γ



 |y − x|−2 dX0 (y) .

(b) There is a c(d) > 0 such that if x ∈ S(X0 ), then as ε ↓ 0 2
|y − x|−2 dX0 (y)(log 1/ε)−1 γ PX0 (X hits B(x, ε)) ∼ c(d)
|y − x|2−d dX0 (y)εd−4 γ

if d = 4 if d > 4.

(c) There is a Kd > 0 so that if d(x, S(X0)) ≥ 2ε0 , then * Kd (log 1/ε)−1 ∀ε ∈ (0, ε0 ∧ ε20 ) X0 (1)ε2−d PX0 (X hits B(x, ε)) ≤ 0 ∀ε ∈ (0, ε0 ) εd−4 γ

if d = 4 if d > 4.

Remark. The constant c(d) is the one arising in Lemma III.5.9 below. Proof. By translation invariance we may assume x = 0. Choose f ∈ Cb (Rd )+ , radially symmetric so that {f > 0} = B(0, 1). Let fε (x) = f (x/ε), and let uλ,ε (t, x) be the unique solution of ∆u γu2 ∂u = − + λfε , u0 = 0. ∂t 2 2

(SE)ε

4

4

By scaling, uλ,ε (t, x) = ε−2 uλε ,1 (tε−2 , xε−1 ) ≡ ε−2 uλε (tε−2 , xε−1 ). We have (III.5.9) t      4 Xs (fε ) ds = exp − ε−2 uλε (tε−2 , xε−1 ) dX0 (x) . PX0 exp −λ 0

The left side is decreasing in t and λ and so by taking X0 = δx we see that uλ (t, x) ↑ u(x) = u(|x|) (≤ ∞) as t, λ ↑ ∞. Take limits in (III.5.9) to get  ∞  Xs (fε ) ds = 0 PX0 (Xs (B(0, ε)) = 0 ∀s > 0) = PX0 0 t    Xs (fε ) ds . = lim PX0 exp − λ λ,t→∞ 0 −2 −1 (III.5.10) = exp{− ε u(xε ) dX0 (x)}.

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The left-hand side increases as ε ↓ 0 and so (take X0 = δx ), (III.5.11)

ε → ε−2 u(x/ε) decreases as ε ↓ 0, and in particular the radial function u(x) is decreasing in |x|.

By taking ε = 1 and X0 = δx in (III.5.10), we see that (III.5.12)

u(x) = − log Pδx (Xs (B(0, 1)) = 0 ∀s > 0).

Suppose |x| > 1 and Pδx (Xs (B(0, 1)) = 0 ∀s > 0) = 0. Then by the multiplicative property (III.5.13)

P n1 δx (Xs (B(0, 1)) = 0 ∀s > 0) = 0 ∀n ∈ N.

The extinction probability formula (II.5.12) and Historical Modulus of Continuity (Theorem III.1.3) imply (δ(3, ω) is as in the latter result) for some C, ρ > 0, Q0, n1 δx (X n1 = 0, δ(3) > 1/n) ≥ e−2/γ −

C1 1 1 ≥ e−2/γ if n ≥ n0 . γ n nρ 2

The above event implies R ⊂ B(x, 3h(1/n)) and so Xs (B(0, 1)) = 0 ∀s > 0 providing 3h(1/n) < |x| − 1. This contradicts (III.5.13) and so we have proved u(x) < ∞ for all |x| > 1, and so is bounded on {|x| ≥ 1+ε} ∀ε > 0 by (III.5.11). Letting λ, t → ∞ in (SE)1 , leads one to believe that u solves (III.5.14)

∆u = γu2 on {|x| > 1}, lim u(x) = ∞, |x|↓1

lim u(x) = 0.

|x|→∞

Lemma III.5.8. As λ, t ↑ ∞, uλ (t, x) ↑ u(x), where for |x| > 1, u is the unique non-negative solution of (III.5.14). Moreover u is C 2 on {|x| > 1}. Proof. The mild form of (SE) (recall (M E)0,f from Section II.5) gives (Bt is a Brownian motion under P x )    t γ  t f (Bs ) ds . (III.5.15) uλt (x) = −E x uλt−s (Bs )2 ds + λE x 0 2 0 / C}. If x ∈ C, Let C be an open ball with C ⊂ {|x| > 1} and let TC = inf{t : Bt ∈ the strong Markov property shows that  t∧TC γ  uλt (x) =E x − uλt−s (Bs )2 ds 2 0   (t−TC )+ γ  − uλ(t−TC )+ −s (Bs )2 ds + E x E B(TC ) 2 0   (t−TC )+ f (Bs ) ds , + λE B(TC ) 0

and so by (III.5.15), with ((t − TC ) , B(TC )) in place of (t, x),   t∧TC γ   uλt−s (Bs )2 ds = E x uλ(t−TC )+ (BTC ) . uλt (x) + E x 2 0 +

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Now let t, λ → ∞ and use Monotone Convergence to see   TC γ x u(Bs )2 ds = E x (u(BTC )) ∀x ∈ C. (III.5.16) u(x) + E 2 0 The
righthand side is harmonic and therefore C 2 on C. The second term on the left is γ2 C gC (x, y)u(y)2 dy, where gC is the Green function for C. This is C 2 on C by Theorem 6.6 of Port-Stone (1978). Itˆ o’s Lemma and (III.5.16) gives  TC ∆  γ  TC u(Bs )2 ds)/E x (TC ). Ex u(Bs ) ds /E x (TC ) = E x 2 2 0 0 Now let C = B(x, 2−n ) ↓ {x} to see ∆u = γu2 on {|x| > 1}. Let µ(t, x) and σ 2 (t, x) be the mean and variance of Xt (B(0, 1)) under Pδx . Then µ(|x| − 1, x) → µ > 0 as |x| ↓ 1 and σ 2 (|x| − 1, x) ≤ γ(|x| − 1) → 0 as |x| ↓ 1, by our moment formulae (Exercise II.5.2). Therefore Lemma III.5.5 and (III.5.12) show that for |x| > 1 e−u(x) = Pδx (Xs (B(0, 1) = 0 ∀s > 0) ≤ 1 − Pδx (X|x|−1 (B(0, 1)) > 0) ≤ σ 2 (|x| − 1, x)/(µ(|x| − 1, x)2 + σ 2 (|x| − 1, x)) → 0 as |x| ↓ 1. Therefore lim|x|↓1 u(x) = ∞. (III.5.11) shows that for |x| ≥ 2, (III.5.17)

u(x) =

4  2 −2  2x 1 2  4 u u(2) → 0 as |x| → ∞. ≤ |x|2 |x| |x| |x| |x|2

It remains only to show uniqueness in (III.5.14). This is an easy application of the classical maximum principle. Let u, v be solutions and c > 1. Then w(x) = c2 u(cx) solves ∆w = γw2 on {|x| > c−1 }. Therefore lim w(x) − v(x) = −∞,

|x|↓1

lim (w − v)(x) = 0

|x|→∞

and the usual elementary argument shows that w − v cannot have a positive local maximum. Therefore w ≤ v and so (let c ↓ 1) u ≤ v. By symmetry u = v. In view of (III.5.10) to obtain the required estimates for Theorem III.5.7 we need to know the asymptotic behavior of u(x) = u(|x|) as |x| → ∞. By the radial symmetry (III.5.14) is an ordinary differential equation in the radial variable and so precise results are known. Lemma III.5.9. As r → ∞

2 −2   γ (4 − d)r 2 −2 u(r) ∼ γ r / log r   c(d) r 2−d , c(d) > 0 γ

d≤3 d=4. d>4

Proof. Iscoe (1988) gives just enough hints to reduce this to a calculus exercise (albeit a lengthy one if d = 4) before referring the reader to the differential equations

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231

literature. We will carry out this exercise later in this Section to give a self-contained proof. We are ready to complete the Proof of Theorem III.5.7. Consider first the proof of (b) and (c) when d = 4 (a similar argument works if d > 4). From (III.5.10), we have (III.5.18)

PX0 (X hits B(0, ε)) = PX0 (Xs (B(0, ε)) > 0 ∃s > 0) −2 = 1 − exp{−ε u(x/ε) dX0 (x)}.

Let 2ε0 = d(0, S(X0)) > 0. If x ∈ S(X0 ) and 0 < ε < ε0 ∧ ε20 , then the monotonicity of u and Lemma III.5.9 show that (III.5.19)

(log 1/ε)ε−2 u(x/ε) ≤ (log 1/ε)ε−2 u(2ε0 /ε) ≤ C(log 1/ε)ε−2 (ε0 /ε)−2 (log(ε0 /ε))−1 ≤ C2ε−2 0 .

This proves the left side is uniformly bounded on S(X0 ) and so (III.5.18), Lemma III.5.9 and Dominated Convergence imply lim(log 1/ε)PX0 (X hits B(0, ε)) = lim (log 1/ε)ε−2 u(x/ε) dX0 (x) ε↓0 ε↓0 = lim(log 1/ε)ε−2 u(x/ε) dX0 (x) ε↓0 2 = |x|−2 dX0 (x). γ This proves (b). To prove (c) use (III.5.18) and then (III.5.19) to see that for ε < ε0 ∧ ε20 PX0 (X hits B(0, ε)) ≤ ε−2 u(x/ε) dX0 (x) −1 . ≤ 2Cε−2 0 X0 (1)(log 1/ε)

For (a) we consider 3 cases. Case 1. 0 ∈ S(X0 ). Let 2ε0 = d(0, S(X0)). By (III.5.11), and as in the proof of (b), if x ∈ S(X0 ) and 0 < ε < ε0 , ε−2 u(x/ε) ≤ ε−2 u(2ε0 /ε) ≤ ε−2 0 u(2). This allows us to use Dominated Convergence and Lemma III.5.9 to let ε ↓ 0 in (III.5.10) and conclude   PX0 ∪ε>0 {Xs (B(0, ε)) = 0 ∀s > 0} (III.5.20)   2 (4 − d)|x|−2 dX0 (x) . = exp − γ The event on the left hand side clearly implies 0 ∈ R. Conversely suppose 0 ∈ R. The historical modulus of continuity and 0 ∈ S(X0 ) imply 0 ∈ R([0, δ]) for some δ > 0 w.p. 1. Therefore 0 ∈ R([0, ∞)) and so d(0, R([0, ∞))) > 0 which implies

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for some ε > 0 Xs (B(0, ε)) = 0 for all s ≥ 0. Therefore (III.5.20) is the required equation. Case 2. 0 ∈ S(X0 ), X0 ({0}) = 0. For η > 0, let X0 = X01,η + X02,η , where dX01,η (x) = 1(|x| > η) dX0 (x). If δ > 0, ¯ ∞))) PX0 (0 ∈ R([δ, ∞))) = PX 1,η (0 ∈ R([δ, ∞)))PX 2,η (0 ∈ R([δ, 0

0

≥ PX 1,η (0 ∈ R)PX 2,η (Xδ = 0) 0 0   −2X 2,η (1)   −2(4 − d) 0 = exp 1(|x| > η)|x|−2 dX0 (x) exp , γ γδ where we have applied Case 1 to X01,η and used the extinction probability formula (II.5.12) in the last line. Let η ↓ 0 and then δ ↓ 0 to get the required lower bound. / R) ≤ PX 1,η (0 ∈ / R), apply case 1 to X01,η , For the upper bound just use PX0 (0 ∈ 0 and let η ↓ 0. Case 3. X0 ({0}) > 0. Note that PX0 (Xδ ({0})) = P X0 (Bδ = 0) = 0 and so we may use the Markov property and apply the previous cases a.s to Xδ to conclude PX0 (X misses {0}) ≤ PX0 (PXδ (Xmisses {0}))    2(4 − d) = PX0 exp − → 0 as δ ↓ 0 |x|−2 dXδ (x) γ because the weak continuity and Fatou’s lemma shows lim inf |x|−2 dXδ (x) = ∞ a.s. δ↓0

Therefore X hits {0} PX0 -a.s. and the result holds in this final case. Corollary III.5.10. Let fβ (r) =

*

rβ (log 1/r)−1

if β > 0 , if β = 0

and d ≥ 4. If A ∈ B(Rd ) and fd−4 − m(A) = 0, then A is polar for X. In particular, points are polar for X. n n Proof. Assume without loss of generality that A is bounded. Let A ⊂ ∪∞ i=1 B(xi , ri ) ∞ n where limn→∞ i=1 fd−4 (2ri ) = 0. Choose x0 ∈ A and n large so that n n d(x0 , ∪∞ 1 B(xi , ri )) = ε0 > 0. Then Theorem III.5.7(c) implies that for n large, Pδx0 (R ∩ A = φ) ≤

∞ 

Pδx0 (R ∩ B(xni , rin ) = φ)

i=1

≤

∞ Kd 2−d  ε0 fd−4 (rin ) γ i=1

→ 0 as n → ∞. Remark. (III.5.1) shows that this result is also a consequence of Theorem III.5.2 as the hypothesis on A is stronger than that in Theorem III.5.2. However, in practice it

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233

is often easier to verify the Hausdorff measure condition than the capacity condition, and (III.5.2)–(III.5.3) show the conditions are close. Exercise III.5.1. Let * + 4 ψ˜d (r) = r 4 log 1/r if d = 4 . if d > 4 r Use Theorem III.5.7(c) to show that ψ˜d − m(R([δ, ∞))) < ∞ ∀δ > 0 PX0 – a.s. Conclude that R is PX0 a.s. Lebesgue null if d ≥ 4. Hint. You may assume without loss of generality that X0 = δ0 . Why? Proof of Lemma III.5.9. By considering v = γu, we may assume without loss of generality that γ = 1. Radial symmetry shows that (III.5.14) is equivalent to the ordinary differential equation u (r) +

(III.5.21)

d−1  u (r) = u(r)2 for r > 1 r

with the associated boundary conditions, or equivalently (III.5.22)

(r d−1 u ) = r d−1 u(r)2 for r > 1,

lim u(r) = 0,

r→∞

lim u(r) = ∞.

r→1+

This shows that r d−1 u is non-decreasing and, as it is non-positive by (III.5.11), we conclude that (III.5.23)

−c0 (d) = lim r d−1 u (r) ≤ 0. r→∞

If u(r0 ) = 0 for some r0 > 1, then (III.5.12) implies that X1 (B(0, 1)) = 0 Pδr0 a.s. This contradicts the fact that Pδr0 (X1 (B(0, 1)) = P r0 (B1 ∈ B(0, 1)) > 0 and we have proved that (III.5.24)

u(r) > 0 for all r > 1.

Integrate (III.5.22) twice, and use (III.5.23) and u(∞) = 0 to obtain the integral equation ∞ ∞  ∞  t1−d dt + t1−d sd−1 u(s)2 ds dt. (III.5.25) u(r) = c0 (d) r

r

t

This shows that c0 (d) = 0 if d ≤ 2 (or else the first term would be infinite), and for d ≥ 3 the above gives ∞  2−d − s2−d  d−1 r c0 (d) 2−d r + s u(s)2 ds. (III.5.26) u(r) = d−2 d − 2 r We claim that (III.5.27)

c0 (d) > 0 iff d ≥ 5.

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0 (d) 2−d r , and therefore for Assume first that d = 3 or 4. (III.5.26) implies u(r) ≥ cd−2  some c > 0, ∞ ∞ c0 (d)2 4−2d  2 2−d r 2−d sd−1 s ds ≥ c c (d) r s3−d ds. u(r) ≥ c 0 (d − 2)2 2r 2r

As the last integral is infinite, this shows c0 (d) must be 0. Assume now that d ≥ 5 and c0 (d) = 0. Then (III.5.26) implies 1 2−d ∞ d−1 (III.5.28) u(r) ≤ r s u(s)2 ds  r 2−d as r → ∞, d−2 r because (III.5.26) implies the above integral is finite. Use this in (III.5.26) to see there is an r0 > 1 such that r ≥ r0 implies 1 3−d ∞ d−1 s u(s)2 ds r ru(r) ≤ d−2 r ∞ su(s)ds ≤ r 3−d r ∞ su(s)ds, ≤ r

where the above integral is finite by (III.5.28). Now iterate the above inequality as in Gronwall’s lemma to see that ru(r) = 0 for r ≥ r0 , contradicting (III.5.24). This completes the proof of (III.5.27). Assume d ≥ 5. Then (III.5.26) implies ∞ 1 c0 (d) r d−2 u(r) = + r d−2 [r 2−d − s2−d ]sd−1 u(s)2 ds d−2 d−2 r c0 (d) + h(r). ≡ d−2
∞ Clearly h(r) ≤ c r sd−1 u(s)2 ds → 0 as r → ∞ because (III.5.26) implies the 0 (d) . above integrals are finite. This proves the required result with c(d) = cd−2 Assume d ≤ 4 so that (III.5.25) and (III.5.27) give ∞ ∞ t1−d sd−5 (s2 u(s))2 dsdt. (III.5.29) u(r) = r

t

Recall from (III.5.11) that r u(r) ↓ L ≥ 0 as r → ∞. Assume d ≤ 3 and L = 0. If ε ∈ (0, 1), there is an r0 (ε) > 1 so that r 2 u(r) ≤ ε whenever r ≥ r0 . Now use (III.5.29) to see that for r ≥ r0 , ∞ −3 ε2 ε t dt = ≤ . r 2 u(r) ≤ ε2 r 2 4−d (4 − d)2 2 r 2

Iterate the above to see that u(r) = 0 for all r ≥ r0 , which contradicts (III.5.24) and hence proves L > 0. The fact that r 2 u(r) ↓ L and (III.5.29) together imply ∞ ∞ ∞ ∞ t1−d sd−5 dsdt (r 2 u(r))2 ≥ r 2 u(r) ≥ L2 r 2 t1−d sd−5 dsdt, r2 r

t

r

t

Superprocesses and therefore

235

L2 1 (r 2 u(r))2 ≥ r 2 u(r) ≥ . 2(4 − d) 2(4 − d) 2

L . This implies that L = 2(4 − d) (because L > 0) Let r → ∞ to see that L = 2(4−d) and so the result follows for d ≤ 3. It remains to consider the 4-dimensional case which appears to be the most delicate. In this case

w(r) ≡ r 2 u(r) ↓ L = 0

(III.5.30)

because if L were positive the inner integral in (III.5.29) would be infinite. (III.5.26) shows that (recall c0 (4) = 0) 1 ∞ 1 ∞ 3 s u(s)2 ds − su(s)2 dsr 2 w(r) = 2 r 2 r ∞  1 ∞ −1 = s w(s)2 ds − s−3 w(s)2 dsr 2 2 r r  1 ∞ −1 2 ≡ (III.5.31) s w(s) ds − g(r) . 2 r The monotonicity of w shows that (III.5.32) g(r) ≤ w(r)2

∞

s−3 dsr 2 = w(r)2 /2 ↓ 0 as r → ∞.

r

Let

∞ 1 w(s)2 1 log r ds − (log r)g(r) by (III.5.31) 2 s 2 r 1 1 ≡ h(r) − (log r)g(r) 2 2

v(r) ≡ (log r)w(r) = (III.5.33) Note that by (III.5.32),

1 (log r)g(r) 2

(III.5.34)

v(r)

=

1 g(r) 1 ≤ w(r) → 0 as r → ∞, 2 w(r) 4

and so (III.5.35) Now

lim

r→∞

1 h(r) 2

v(r)

= 1.

1 ∞ w(s)2 (log r)w(r)2 ds − r r s r  1

(by (III.5.31)) = 2w(r) + g(r) − (log r)w(r)2 r  w(r)

g(r) 1 1 = 2+ − h(r) + (log r)g(r) . r w(r) 2 2

h (r) =

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Superprocesses

(III.5.34) and (III.5.35) imply 12 (log r)g(r) = ε(r)h(r), where limr→∞ ε(r) = 0, and g(r) (III.5.32) shows that d(r) = w(r) → 0 as r → ∞. We can therefore rewrite the above as (III.5.36)

h (r) =

w(r) a(r)[b(r) − h(r)], r

where limr→∞ a(r) = 12 , and limr→∞ b(r) = 4. We claim that limr→∞ h(r) exists in [0, ∞). If h(r) > 4 for large enough r this is clear since h is eventually decreasing (by (III.5.36)) and bounded below. In a similar way the claim holds if h(r) < 4 for large enough r. Assume therefore that h(r) ≤ 4 for some arbitrarily large r and h(r) ≥ 4 for some arbitrarily large values of r. We claim that limr→∞ h(r) = 4. Let ε > 0 and suppose lim supr→∞ h(r) > 4 + ε. We may choose rn ↑ ∞ and sn ∈ (rn , rn+1 ) so that h(rn ) ≥ 4 + ε and h(sn ) ≤ 4 and then choose un ∈ [sn−1 , sn ] so that h has a local maximum at un and h(un ) ≥ 4 + ε. This implies h (un ) = 0 which contradicts (III.5.36) for n sufficiently large. We have proved that lim supr→∞ h(r) ≤ 4. A similar argument shows that lim inf r→∞ h(r) ≥ 4. In this way the claim is established. This together with (III.5.33) and (III.5.34) shows that (III.5.37)

L = lim v(r) exists in R+ . r→∞

An argument similar to that for d ≤ 3, and using (III.5.29), shows that L > 0. We can write (III.5.33) as ∞ v(s)2 1 ds − ε(r), (III.5.38) v(r) = log r 2 (log s)2 s r where limr→∞ ε(r) = 0 (ε(r) ≥ 0) by (III.5.34) and (III.5.37). If L > ε > 0, there is an r0 > 1 such that for r ≥ r0 , ∞ ∞ 1 1 (log r)(L−ε)2 (log s)−2 s−1 ds−ε ≤ v(r) ≤ (log r)(L+ε)2 (log s)−2 s−1 ds. 2 2 r r Let r → ∞ and then ε ↓ 0 to see that L = 12 L2 and so L = 2. The result for d = 4 follows. We next consider the fixed time analogue of Theorem III.5.7. Let ∆ 2 continue to denote the generator of Brownian motion. Recall from Theorem II.5.9 and Example II.2.4(a) that if φ ∈ Cb2 (Rd )+ and Vt φ ≥ 0 is the unique solution of (SE)φ,0

∆Vt γ ∂V = − Vt2 , ∂t 2 2

V0 = φ,

then (LE)

PX0 (exp(Xt (φ))) = exp(−X0 (Vt φ)).

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237

Theorem III.5.11. Let d ≥ 3. There is a constant Cd such that for all X0 ∈ MF (Rd ), all t ≥ ε2 > 0, and all x ∈ Rd , Cd PX0 (Xt (B(x, ε)) > 0) ≤ pε2 +t (y − x)X0 (dy)εd−2 γ Cd −d/2 ≤ X0 (Rd )εd−2 . t γ Proof. Since Xγt is a SBM(1) starting at X0 /γ (check the martingale problem as in Exercise II.5.5) we clearly may assume γ = 1. By translation invariance we may assume x = 0. Let φ ∈ Cb2 (Rd )+ be a radially symmetric function such that {φ > 0} = B(0, 1), let φε (x) = φ(x/ε) and let v λ,ε (t, x) ≥ 0 be the unique solution of (SE)λφε ,0 from Theorem II.5.11. By scaling we have v λ,ε (t, x) = ε−2 v λε

2

,1

2

(tε−2 , xε−1 ) ≡ ε−2 v λε (tε−2 , xε−1 ).

By (LE),

 2 (III.5.39) 1 −PX0 (exp(−λXt (φε ))) = 1 −exp − ε−2 v λε (tε−2 , xε−1 )X0 (dx) . The left-hand side is increasing in λ, and so by taking X0 = δx we see that v λ (t, x) ↑ v ∞ (t, x) ≤ ∞. Let λ → ∞ in (III.5.39) to conclude that  PX0 (Xt (B(0, ε)) > 0) = 1 − exp − ε−2 v ∞ (tε−2 , xε−1 )X0 (dx) (III.5.40) (III.5.41) ≤ ε−2 v ∞ (tε−2 , xε−1 )X0 (dx). We therefore require a good upper bound on v ∞ . A comparison of (III.5.40), with X0 = δx , ε = 1, and the extinction probability (II.5.12) shows that (III.5.42)

v ∞ (t, x) ≤

2 . t

To get a better bound for small t and |x| > 1 we let r > 1 and suppose (III.5.43)

there exist tn ↓ 0 such that sup v ∞ (tn , x) → ∞. |x|≥r

Then (III.5.40) implies that limn→∞ Pδxn (Xtn (B(0, 1)) > 0) = 1 for some |xn | ≥ r. Therefore by translation invariance, lim inf Pδ0 (Xtn (B(0, r − 1)c ) > 0) = lim inf Pδxn (Xtn (B(xn , r − 1)c ) > 0) n→∞

n→∞

≥ lim inf Pδxn (Xtn (B(0, 1)) > 0) = 1. n→∞

On the other hand our historical modulus of continuity (recall Corollary III.1.5) shows the left-hand side of the above is 0. Therefore (III.5.43) must be false. This

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Superprocesses

with (III.5.42) proves that (III.5.44)

M (r) ≡

sup

|x|≥r,t>0

v ∞ (t, x) < ∞

∀r > 1.

Proposition II.5.12 gives the Feynman-Kac representation   −1 t  v λ (t − s, Bs ) ds . (III.5.45) v λ (t, x) = E x λφ(Bt ) exp 2 0 Use the strong Markov property at Tr = inf{t : |Bt | = r} to see that (III.5.45) implies that if |x| ≥ r > 1, then  −1 Tr   v λ (t, x) =E x 1(Tr < t) exp v λ (t − s, Bs ) ds 2 0   −1 t−Tr  × E BTr λφ(Bt−Tr ) exp v λ (t − Tr − s, Bs ) ds 2 0 ≤E x (1(Tr < t)v λ (t − Tr , BTr )). Let λ → ∞ and use Monotone Convergence in the above to see that (III.5.46)

v ∞ (t, x) ≤ E x (1(Tr < t)v ∞ (t − Tr , BTr )), |x| ≥ r > 1.

If we replace Tr by the deterministic time t − s (0 ≤ s ≤ t) in the above argument we get (III.5.47)

v ∞ (t, x) ≤ Pt−s (v ∞ (s, ·))(x).

Combine (III.5.46) with (III.5.44) to see that for |x| ≥ 7 > r = 2, v ∞ (1, x) ≤ M (2)P x (T2 < 1) ≤ 2M (2)P 0 (|B1 | > |x| − 2) (by a d-dimensional reflection principle) ≤ c1 exp(−|x|2 /4), where we used our bound |x| > 7 in the last line. Together with (III.5.42), this gives v ∞ (1, x) ≤ c2 p(2, x) for all x ∈ Rd , and so (III.5.47) with s = 1 implies v ∞ (t, x) ≤ c2 p(t + 1, x) for all t ≥ 1 and x ∈ Rd . Use this in (III.5.41) to conclude that for t ≥ ε2 , −2 p(tε−2 + 1, xε−1 )X0 (dx) PX0 (Xt (B(0, ε)) > 0) ≤ c2 ε = c2 εd−2 p(t + ε2 , x)X0 (dx). This gives the first inequality and the second inequality is then immediate. A corresponding lower bound is now left as an exercise.

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239

Exercise III.5.2. Use Lemma III.5.5 and our first and second moment formulae (Exercise II.5.2) to prove: (a) If d ≥ 3, for any K ∈ N, δ > 0 ∃ε0 (δ, K) > 0 and a universal constant cd > 0 so that whenever X0γ(1) ≤ K, cd PX0 (Xt (B(x, ε)) > 0) ≥ pt (y − x)X0 (dy)εd−2 ∀0 < ε < ε0 (δ, K), t ≥ δ. γ (b) If d = 2, show the conclusion of (a) holds with (log 1/ε)−1 in place of εd−2 and the additional restriction t ≤ δ −1 . Hints: (1) Draw a picture or two to convince yourself that P y (Bt ∈ B(x, ε)) ≥ cd εd pt (y − x) (2) If B = B(0, ε), then Ps ((Pt−s 1B )2 ) ≤ cd



εd (t−s)d/2

∀t ≥ ε2 .  ∧ 1 Pt 1 B .

Remark III.5.12. (a) If d ≥ 3, then Theorem III.5.11 and the above Exercise give sharp bounds on PX0 (Xt (B(x, ε)) > 0 as ε ↓ 0 except for the value of the constants Cd and cd . Theorem 3.1 of Dawson-Iscoe-Perkins (1989) shows that there is a universal constant c(d) > 0 such that (III.5.48) lim ε2−d PX0 (Xt (B(x, ε)) > 0) = c(d) pt (y − x)X0 (dy). ε↓0

If d = 2, a companion upper bound to Example III.5.2(b) PX0 (Xt (B(x, ε)) > 0) ≤ C2 t−1 X0 (1)| log ε|−1

∀t ∈ [ε, ε−1 ] ∀ε ∈ (0, 1/2)

is implied by Corollary 3 of Le Gall (1994). A version of (III.5.48) has not been obtained in this more delicate case. (b) The analogue of Theorem III.5.2 for fixed times is (III.5.49) A ∩ S(Xt ) = ∅ PX0 − a.s. iff C(gd−2 )(A) = 0, ∀A ∈ B(Rd ), t > 0, d ≥ 1, X0 = 0. The reader may easily prove the necessity of the capacity condition for fixed time polarity by means of a straightforward adaptation of the proof of Theorem III.5.4. This result was first proved in Perkins (1990) (see Theorem 6.1). As in Corollary III.5.10, Theorem III.5.11, and Le Gall’s companion upper bound if d = 2, readily show that for d ≥ 2, and all t > 0 and A ∈ B(Rd ), (III.5.50)

fd−2 − m(A) = 0 ⇒ A ∩ S(Xt ) = ∅ PX0 − a.s. .

This is of course weaker than the sufficiency of the capacity condition in (III.5.49) which is again due to Dynkin (1992). Parts (a) and (b) of the following Exercise will be used in the next Section. Exercise III.5.3. Let X 1 , X 2 be independent SBM’s with branching rate γ which start at X01 , X02 ∈ MF (Rd ) − {0}. (a) If d ≥ 5 use Theorem III.5.11 to show that S(Xt1 ) ∩ S(Xt2 ) = ∅ a.s. (b) Prove that the conclusion of (a) remains valid if d = 4.

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Superprocesses

Hint. One way to handle this critical case is to first apply Theorem III.3.8 (which has not been proved in these notes) to X 1 . (c) If d ≤ 3 show that P (S(Xt1 ) ∩ S(Xt2 ) = ∅) > 0. Hint. Use the necessity of the capacity condition in (III.5.49) together with (III.5.2) (the connection between capacity and Hausdorff measure) and Corollary III.3.5 (the latter results are only needed if d > 1). 6. Disconnectedness of the Support Our goal in this Section is to study the disconnectedness properties of the support of super-Brownian motion. The results seem to be scattered in the literature, often with proofs that are only sketched, and we will try to collect them in this section and give a careful proof of the main result (Theorem 6.3). This also gives us the opportunity to advertise an intriguing open problem. The historical clusters of Theorem III.1.1 will be used to disconnect the support and we start by refining that result. Assume we are in the setting of Section II.8: H is the (Y, γ, 0)-historical process on the canonical space of paths ΩH [τ, ∞) with law Qτ,m , and γ is a positive constant. Recall the canonical measures {Rτ,t(y, ·) : t > τ, y ∈ Dτ } associated with H from (II.8.6) and set ∗ Pτ,t (y, A) =

Rτ,t (y, A) γ(t − τ ) = Rτ,t (y, A). Rτ,t (y, MF (D) − {0}) 2

∗ as the law of Ht starting from an inFrom Exercise II.7.2 we may interpret Pτ,t finitesimal point mass on y at time τ and conditioned on non-extinction at time t. If ν is a probability on MFτ (D) we abuse our notation slightly and write Qτ,ν for
Qτ,m ν(dm) and also adopt a similar convention for the laws Pµ (µ a probability on MF (E)) of the corresponding superprocess. If τ ≤ s ≤ t, define

rs,t (Ht ) ∈ MF (Ds × D) by rs,t (Ht )(A) = Ht ({y : (y s , y) ∈ A}). Theorem III.6.1. Let m ∈ MFτ (D) − {0}, t > τ and let Ξ be a Poisson point process on Dτ × ΩH [t, ∞) with intensity 2 ∗ (y,·) (B) m(dy). µ(A × B) = 1A (y)Qt,Pτ,t γ(t − τ ) Then (under Qτ,m ) D
(a) rτ,t (Ht )≡ δy × νt Ξ(dy, dν),   D
(b) (Hu )u≥t ≡ νu Ξ(dy, dν) . u≥t

Proof. (a) Let A1 , . . . , An be a Borel partition of D, let φi : D → R+ be bounded Borel maps i = 1, . . . , n, and define Hti (·) = Ht (· ∩ {y τ ∈ Ai }) and f : Dτ × D → R+

Superprocesses

241

n

1Ai (y)φi (w). Then (III.1.3) and (II.8.6)(b) imply Qτ,m ( exp(− f (y τ , y)Ht(dy))    n  i Ht (φi ) = Qτ,m exp −

by f (y, w) =

i=1

i=1

=

n , i=1

(III.6.1)

Qτ,m (exp(−Hti (φi ))) 

= exp −

n  i=1

−ν(φi )

1−e

 2 ∗ P (y, dν)1Ai (y)dm(y) . γ(t − τ ) τ,t

On the other hand   E exp − f (y, w)νt(dw)Ξ(dy, dν)  = E(exp − (1 − exp(− f (y, w)νt(dw)))µ(dy, dν)   n  2 ∗ 1Ai (y)νt (φi )) = exp − 1 − exp(− P (y, dνt )m(dy) γ(t − τ ) τ,t i=1   n  2 −ν(φi ) ∗ P (y, dν)1Ai (y)dm(y) . (III.6.2) = exp − 1−e γ(t − τ ) τ,t i=1 As (III.6.1) and (III.6.2) are equal, the two random measures in (a) are equal in law by Lemma II.5.9. 2m(D) (b) Let {yi , H i ) : i ≤ N } be the points of Ξ. Then N is Poisson with mean γ(t−τ ), given N , {yi : i ≤ N } are i.i.d with law m/m(D), and (III.6.3)

given σ(N, (yi )i≤N ), {H i : i ≤ N } are independent random ∗ (y ,·) . processes with H i having law Qt,Pτ,t i

¯ · = N H i is a sum of N inTherefore conditional on σ(N, (yi )i≤N , (Hti )i≤N ), H · i=1 dependent historical processes. Such a sum will clearly satisfy (HM P )t,H¯ t (ω) and therefore be a historical process itself (this is the multiplicative property of super¯ u )u≥t given σ(N, (yi )i≤N , (H i )i≤N ) processes). Therefore the conditional law of (H t ¯ is Qt,H¯ t . Part (a) implies that Ht has law Qτ,m (Ht ∈ ·). The Markov property of ¯ u≥t has law Qτ,m ((Hu )u≥t ∈ ·) which is what we H under Qτ,m now shows that (H) have to prove. We now reinterpret the above result directly in terms of the historical process H on its canonical space (ΩH [τ, ∞), Qτ,m). We in fact assume that H is the historical process of a super-Brownian motion with constant branching rate γ, although the result and its proof remain valid for any historical process such that Hs has no atoms Qτ,m -a.s. for any s > τ . If τ ≤ s < t and y ∈ Ds , let Hts,y (·) = Ht ({w ∈ · : ws = y}), i.e., Hts,y is the contribution to Ht from descendants of y at time s.

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Corollary III.6.2. Let m ∈ MFτ (D) and τ ≤ s < t. Assume either τ < s or m is non-atomic. Then under Qτ,m : (a) Conditional on F H [τ, s+], S(rs (Ht )) is the range of a Poisson point process with s (ω) . intensity 2H γ(t−s) (b) Hu = y∈S(rs (Ht )) Hus,y for all u ≥ t a.s. (c) Conditional on F H [τ, s+] ∨ σ(S(rs (Ht ))), {(Hus,y )u≥t : y ∈ S(rs (Ht ))} are independent processes and for each y ∈ S(rs (Ht )), (Hus,y )u≥t has (conditional) law ∗ (y,·) . Qt,Ps,t Proof. (a) is included in Theorem III.1.1. (b) Lemma III.1.2 and the Markov property show that Hu (S(rs (Ht ))c )u≥t is a continuous martingale starting at 0 and so is identically 0 a.s. (b) now follows from the definition of Hus,y . (c) Theorem III.3.4 and Exercise II.8.3 show that Hs is non-atomic a.s (use Theorem III.3.8(c) if d = 1). Therefore by the Markov property we may assume without loss of generality that s = τ and m is non-atomic. We must show that conditional on S(rτ (Ht )), {(Huτ,y )u≥t : y ∈ S(rτ (Ht ))} are independent, and, for each y ∈ S(rτ (Ht )), (Huτ,y )u≥t has (conditional) law ∗ (y,·) . As this will depend only on the law of (Hu )u≥t , by Theorem III.6.1(b) Qt,Pτ,t we may assume that Ξ is as in that result and (III.6.4) Hu = νu Ξ(dy, dν) for all u ≥ t. Let {(yi , H i ) : i ≤ N } be the points of Ξ (as in the proof of Theorem III.6.1(b)) so that (III.6.4) may be restated as (III.6.5)

Hu =

N 

Hui for all u ≥ t.

i=1

Theorem III.6.1(a) implies that {yi : i ≤ N } = S(rτ (Ht )) and the fact that m is non-atomic means that all the yi ’s are distinct a.s. By (II.8.6)(a), ν(y τ = yi ) = 0 ∗ (yi , ·)-a.a. ν and so Hti ({y : y τ = yi }) = 0 a.s. As in the proof of (b) we may Pτ,t conclude that Hui ({y : y τ = yi }) = 0 for all u ≥ t a.s. This shows that Hui (·) = Hui (·, y τ = yi ) for all u ≥ t a.s. = Hu (·, y τ = yi ) for all u ≥ t a.s. (by (III.6.5) and yi = yj if i = j a.s.) = Huτ,yi (·) for all u ≥ t a.s. The required result now follows from (III.6.3). We are ready for the main result of this section (a sketch of this proof was given in Perkins (1995b)). In the rest of this Section we assume X, (Pµ )µ∈MF (E) is  a super-Brownian motion with constant branching rate γ and H, (Qτ,m )(τ,m)∈Eˆ is the corresponding historical process. Theorem III.6.3. If d ≥ 4, S(Xt ) is totally disconnected PX0 -a.s. for each t > 0.

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243

˜ Proof. By Exercise II.5.1 (or II.5.3) we may work with Xt = Π(H) t under Q0,X0 . s,y If t > 0 is fixed, 0 ≤ s < t and y ∈ S(rs (Ht )), let Xt (A) = Hts,y (yt ∈ A) ≡ ˆ −1 ˆ Hts,y ◦ Π t (A), where Πt is the obvious projection map. Let εn ↓ 0, where 0 < εn < t. Corollary III.6.2(b) implies (III.6.6)

S(Xt ) = ∪y∈S(rt−εn (Ht )) S(Xtt−εn ,y ) ∀n a.s.

By Corollary III.6.2(c) and (II.8.7), conditional on F H ([0, t−εn ))∨σ(S(rt−εn (Ht ))), {Xtt−εn ,y : y ∈ S(rt−εn (Ht ))} are independent and Xtt−εn ,y has (conditional) law ∗ ˆ −1 ∈ ·) = Rεn (yt−εn , ·) . Pt−ε (y, ν ◦ Π t n ,t Rεn (yt−εn , 1)

A Poisson superposition of independent copies of Xtt−εn ,y has law Pyt−εn (Xεn ∈ ·) by (II.7.11) and so Exercise III.5.3 shows that (III.6.7)

{S(Xtt−εn ,y ) : y ∈ St−εn (Ht )} are disjoint for all n a.s.

Let δ(3, ω) > 0 Q0,X0 -a.s. be as in the Historical Modulus of Continuity (Theorem III.1.3). Then that result and Corollary III.6.2(b) show that S(Htt−εn ,y ) ⊂ S(Ht ) ⊂ K(δ(3, ω), 3) and so by the definition of Xtt−εn ,y , (III.6.8)

S(Xtt−εn ,y ) ⊂ B(yt−εn , 3h(εn )) if εn ≤ δ(3, ω).

Fix ω outside a null set so that (III.6.6), (III.6.7), and (III.6.8) hold, and δ(3, ω) > 0. Then S(Xt ) can be written as the disjoint union of a finite number of closed sets of arbitrarily small diameter and hence is totally disconnected. X02 ,

If X 1 and X 2 are independent super-Brownian motions starting from X01 and respectively, then (see Theorem IV.3.2(b) below)

(III.6.9)

if d ≥ 6, S(Xu1 ) ∩ S(Xu2 ) = ∅ for all u > 0 a.s.

Using this in place of Exercise III.5.3 in the above proof we get a version of the above result which holds for all times simultaneously. Theorem III.6.4. If d ≥ 6, then S(Xt ) is totally disconnected for all t > 0 PX0 − a.s. Proof. Our setting and notation is that of the previous argument. Corollary III.6.2 implies that (III.6.10) S(Xu ) = ∪y∈S(r(j−1)εn (Hjεn )) S(Xu(j−1)εn ,y ) ∀u ∈ [(jεn , (j + 1)εn ] ∀j, n ∈ N a.s. Corollary III.6.2(c), Exercise II.8.3 and (II.8.7) show that, conditional on F H ([0, (j − 1)εn ]) ∨ σ(S(r(j−1)εn (Hjεn ))),

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(j−1)ε ,y

n )u≥jεn : y ∈ S(r(j−1)εn (Hjεn )} are independent and (Xu+jεn )u≥0 has {(Xu law Pµn (y(j−1)εn ) where µn (x, ·) = Rεn (x, ·)/Rεn (x, 1). It therefore follows from (III.6.9) that

(III.6.11)

{S(Xu(j−1)εn ,y ) : y ∈ S(r(j−1)εn (Hjεn ))} are disjoint for all u ∈ (jεn , (j + 1)εn ] ∀j, n ∈ N a.s.

Finally use the Historical Modulus of Continuity and Corollary III.6.2(b), as in the proof of Theorem III.6.3 to see that (III.6.12) S(Xu(j−1)εn ,y ) ⊂ B(y(j−1)εn , 6h(εn )) forall u ∈ [jεn , (j + 1)εn ] ∀j ∈ N and n such that 2εn < δ(3, ω). As before (III.6.10)-(III.6.12) show that with probability one, for all u > 0, S(Xu ) may be written as a finite disjoint union of closed sets of arbitrarily small diameter and so is a.s. totally disconnected. In one spatial dimension the existence of a jointly continuous density for X (see Section III.3.4) shows that the closed support cannot be totally disconnected for any positive time with probability one. This leaves the Open Problem. In two or three dimensions, is the support of super-Brownian motion a.s. totally disconnected at a fixed time? Nothing seems to be known in two dimensions and the only result in this direction for three dimensions is Theorem 6.5. (Tribe (1991)) Let Comp(x) denote the connected component of S(Xt ) containing x. If d ≥ 3, then Comp(x) = {x} for Xt -a.a. x PX0 -a.s. for each t > 0. Tribe’s result leaves open the possibility that there is a non-trivial connected component in S(Xt ) having mass 0. The proof considers the history of a particle x chosen according to Xt and decomposes the support at time t into the cousins which break off from this trajectory in [t − ε, t] and the rest of the population. He then shows that with positive probability these sets can be separated by an annulus centered at x. By taking a sequence εn ↓ 0 and using a zero-one law he is then able to disconnect x from the rest of the support a.s. The status of Theorem 6.5 in two dimensions remains unresolved. The critical dimension for Theorem III.6.4, i.e., above which the support is totally disconnected for all positive times, is also not known. 7. The Support Process In this section we give a brief survey of some of the properties of the set-valued process S(Xt ). Let K be the set of compact subsets of Rd . For non-empty K1 , K2 ∈ K, let ρ1 (K1 , K2 ) = sup d(x, K2 ) ∧ 1, x∈K1

ρ(K1 , K2 ) = ρ1 (K1 , K2 ) + ρ1 (K2 , K1 ), and set ρ(K, φ) = 1 if K = φ. ρ is the Hausdorff metric on K and (K, ρ) is a Polish space (see Dugundji (1966), p. 205,253).

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245

Assume X is SBM(γ) under PX0 and let St = S(Xt ), t ≥ 0. By Corollary III.1.4, {St : t > 0} takes values in K a.s. Although the support map S(·) is not continuous on MF (Rd ), an elementary consequence of the weak continuity of X is that (III.7.1)

lim ρ1 (Ss , St ) = 0

t→s

∀s > 0

a.s.

On the other hand the Historical Modulus of Continuity (see Corollary III.1.5) shows that if 0 < t − s < δ(ω, 3) ρ1 (St , Ss ) = sup d(x, Ss ) ∧ 1 ≤ 3h(t − s) x∈St

and so (III.7.2)

lim ρ1 (St , Ss ) = 0 ∀s > 0 t↓s

a.s.

(III.7.1) and (III.7.2) show that {St : t > 0} is a.s. right-continuous in K. The a.s. existence of left limits is immediate from Corollary III.1.5 and the following simple deterministic result (see Lemma 4.1 of Perkins (1990)): If f : (0, ∞) → K is such that ∀ε > 0 ∃δ > 0 so that   0 ≤ t − u < δ implies f (u) ⊂ f (t)ε ≡ {x : d x, f (t) < ε}, then f possesses left and right limits at all t > 0. (III.7.1) shows that Ss ⊂ Ss− for all s > 0 a.s. When an “isolated colony” becomes extinct at time s at location F one expect F ∈ Ss− − Ss . These extinction points are the only kind of discontinuities which arise. Theorem 4.6 of Perkins (1990) shows that card(St− − St ) = 0 or

1 for all t > 0

a.s.

The nonstandard proof given there may easily be translated into one use the historical process. For d ≥ 3 the countable space-time locations of these extinction points are dense in the closed graph of X, G0 (X) = {(t, x) : x ∈ St , t ≥ 0} = {(t, x) : x ∈ St− , t > 0} ∪ {0} × S0 (see Theorem 4.8 of Perkins (1990)). Of course if S0 ∈ K, the above arguments show that {St : t ≥ 0} is cadlag in K a.s. Assume now S(X0 ) ∈ K. Theorem III.3.8 suggests that in 2 or more dimensions the study of the measure-valued process X reduces to the study of the K-valued process St = S(Xt ), as Xt is uniformly distributed over St according to a deterministic Hausdorff measure at least for Lebesgue a.a. t a.s. If F = {S ∈ K : hd − m(S) < ∞}, then, as one can use finite unions of “rational balls” in the definition of hd − m on compact sets, it is clear that F is a Borel subset of K and Ψ : F → MF (Rd ), given by Ψ(S)(A) = hd − m(S ∩ A), is Borel. The support mapping S(·) is also a Borel

246

Superprocesses

function from MFK (Rd ), the set of measures with compact support, to K. Theorem III.3.8 implies that for s, t > 0 and A ∈ B(F )   PX0 St+s ∈ A | FtX = PXt (Ss ∈ A) = PΨ(St ) (Ss ∈ A) and so {St : t > 0} is a cadlag F -valued Markov process. This approach however, does not yield the strong Markov property. For this we would need a means of recovering Xt from S(Xt ) that is valid for all t > 0 a.s. and although Theorem III.3.8 (b) comes close in d ≥ 3, its validity for all t > 0 remains unresolved. Another approach to this question was initiated by Tribe (1994). Notation. d ≥ 3 Xtε (A) = |S(Xt )ε ∩ A|ε2−d , A ∈ B(Rd ). Here | · | is Lebesgue measure. The a.s. compactness of S(Xt ) shows Xtε ∈ MF (Rd ) ∀t > 0 a.s. Theorem III.7.1. (Perkins (1994)) Assume d ≥ 3. There is a universal constant c(d) > 0 such that lim Xtε = c(d)Xt ∀t > 0 PX0 -a.s. In fact if φ is a bounded ε↓0

2 then PX0 -a.s. there is an ε0 (ω) > 0 so Lebesgue-integrable function and r < d+2 ε r that sup |Xt (φ) − c(d)Xt (φ)| ≤ ε for 0 < ε < ε0 . t≥ε1/4

Remark. c(d) is the constant given in (III.5.21) below which determines the asymp  totic behaviour of PX0 Xt (B(x, ε)) > 0 as ε ↓ 0. It is now easy to repeat the above reasoning with the above characterization of Xt in place of the Hausdorff measure approach to see that t → S(Xt ) is a Borel strong Markov process with cadlag paths. ε 2−d Notation. Φε : K → MF (Rd ) is given * by Φε (S)(A) = |A ∩ S |ε . lim Φ (S) if it exists . Define Φ : K → MF (Rd ) by Φ(S) = n→∞ 1/n 0 otherwise   E = {S  ∈ K : S Φ(S  ) = S  }; ΩE = D([0, ∞), E) with its Borel σ-field F E , canonical filtration FtE , and coordinate maps St . It is easy to check that Φε is Borel and hence so are Φ and E. If d ≥ 3, Theorem III.7.1 implies S(Xt ) ∈ E ∀t > 0 PX0 -a.s. and so for S  ∈ E we may define a probability QS  on ΩE by   QS  (A) = PΦ(S  ) S(X· ) ∈ A .

Corollary III.7.2. Assume d ≥ 3. (ΩE , F E , FtE , St , QS  ) is a Borel strong Markov process with right-continuous E-valued paths. Proof. See Theorem 1.4 of Perkins (1994). Note. At a jump time of S(Xt ) the left limit of the support process will not be in E because S(Φ(S(X)t−)) = S(Xt ) = S(X)t− . Open Problem. Is S(Xt ) strong Markov for d = 2? The potential difficulty here is that S(Xt ) could fold back onto itself on a set of positive Xt measure at an exceptional time t(ω).

IV. Interactive Drifts 1. Dawson’s Girsanov Theorem Our objective is to study measure-valued diffusions which locally behave like DW-superprocesses, much in the same way as solutions to Itˆ o’s stochastic differential equations behave locally like Brownian motions. This means that we want to consider processes in which the branching rate, γ, the spatial generator, A, and the drift, g, all depend on the current state of the system, Xt , or more generally on the past behaviour of the system, X|[0,t]. One suspects that these dependencies are listed roughly in decreasing order of difficulty. In this Chapter we present a general result of Dawson which, for a large class of interactive drifts, will give an explicit formula for the Radon-Nikodym derivative of law of the interactive model with respect to that of a driftless DW-superprocess. We will illustrate these techniques with a stochastic model for two competing populations and hence work in a bivariate setting for most of the time. The models will also illustrate the limitations of the method as the interactions become singular. These singular interactions will be studied in the next Section. Let Ei , Yi , Ai , and γi , i = 1, 2 each be as in Theorem II.5.1 and set Ei = B(Ei ). Let ΩiX = C(R+ , MF (Ei )) with its Borel σ-field F X,i and canonical filtration FtX,i and introduce the canonical space for our interacting populations,     (Ω2 , F 2 , Ft2 ) = Ω1X × Ω2X , F X,1 × F X,2 , F·X,1 × F·X,2 t+ . The coordinate maps on Ω2 will be denoted by X = (X 1 , X 2 ) and P will be the σ-field of (Ft2 )-predictable sets in R+ × Ω2 . For i = 1, 2, let mi ∈ MF (Ei ), and gi denote a P × Ei -measurable map from R+ × Ω2 × Ei to R. A probability P on (Ω2 , F 2 ) will satisfy (M P )m g iff ∀φi ∈ D(Ai )

Xti (φi )

= mi (φi ) +

t

Xsi (Ai φ − gi (s, X, ·)φi)ds + Mti,gi (φi )

0

defines continuous (Ft2 ) − martingales Mti,gi (φ) (i = 1, 2) under P such that t i,gi i,gi j,gj M0 (φi ) = 0 and M (φi ), M (φj )t = δij Xsi (γi φ2i )ds. 0

Implicit in (M P ) is the fact that We have inserted a negative sign in involve a negative drift.


t

Xsi (|gi (s, X, ·)|)ds < ∞ for all 0 front of gi only because our main

t > 0 P-a.s. example will

Example IV.1.1. (Competing Species) Take Ei = Rd , Ai = ∆/2, γi ≡ 1. If pt denotes the Brownian density, ε > 0 and λi ≥ 0, let giε (x, µ) = λi pε (x − y)µ(dy). Consider two branching particle systems, X N = (X 1,N , X 2,N ), as in Section II.3 with independent spatial (Brownian) motions and independent critical binary branching mechanisms but with one change. At t = i/N a potential parent of a 1-particle 2,N )/N . Simlocated at x1 dies before it can reproduce with probability g1ε (x1 , Xt− ilarly a potential parent in the 2-population located at x2 dies with probability

248

Superprocesses

2,N )/N before reaching child-bearing age. This means that the effective g2ε (x2 , Xt− branching distribution for the i population is

1 1 j,N j,N )/N ) + δ2 (1 − giε (xi , Xt− )/N ) (j = i) ν i,N (xi , X N ) = δ0 (1 + giε (xi , Xt− 2 2 N as well as the location and so depends on the current state of the population Xt−
i,N j,N N ε of the parent. Note that kν (xi , Xt− )(dk) = 1 − gi (xi , Xt− )/N (j = i) and so ε N gi (xi , Xt− ) plays the role of gN in (II.3.1). The two populations are competing for resources and so a high density of 1’s near a 2-particle decreases the likelihood of the successful reproduction of the 2 and a high density of 2’s has similar detrimental effect on a 1-particle. λi is the √ susceptibility of the ith population and ε is the range of the interaction. The method of Section II.4 will show that if X0i = mi is the initial measure of the ith population, then {X N } is tight in D(R+ , MF (Rd )2 ) and all limit points are in Ω2 and satisfy t ∀φi ∈ D(∆/2) Xti (φi ) = mi (φi ) + Xsi (∆φi /2)ds 0 t ε ε,λ (CS)m gi (xi , Xsj )φi (xi )Xsi (dxi )ds + Mti (φi ) (i = 1, 2, j = i), − 0

where Mti (φi ) are continuous (Ft2 ) − martingales such that t i i j Xsi (φ2i )ds. M0 = 0 and M (φi ), M (φj )t = δij 0

The only technical point concerns the uniform (in N ) bound required on E(Xti,N (φ)) in the analogue of Lemma II.3.3. However, it is easy to couple X N with branching particle systems with λi = 0, Z i,N , i = 1, 2, (ignore the interactive killing) so that X i,N ≤ Z i,N and so the required bound is immediate from the λi = 0 case. Clearly m (CS)ε,λ m is a special case of (M P )g with gi (s, X, x) = giε (x, Xsj ), j = i. 0 First consider (M P )m g in what should be a trivial case: gi (s, X, x) = gi (x) for i,g 0

i some gi0 ∈ Cb (Ei ). We let (M P )m g 0 denote this martingale problem. Let Pmi be the 0 0 law of the (Ai , γi , gi )-DW-superprocess starting at mi . If gi ≡ θi is constant, write

1,g 0

2,g 0

i i,0 m 1 2 i Pi,θ mi for this law and write Pmi for Pmi . Clearly Pm1 × Pm2 satisfies (M P )g 0 but it remains to show that it is the only solution. It is easy to extend the Laplace function equation approach in Section II.5 (see Exercise IV.1.1 below) but another approach is to use the following result which has a number of other interesting applications.

Theorem IV.1.2. (Predictable Representation Property). Let Pm be the law of the (Y, γ, g)-DW-superprocess starting at m on the canonical space of MF (E)-

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249

valued paths (ΩX , FX , FtX ). If V ∈ L2 (FX , Pm ), there is an f in L2 = {f : R+ × ΩX × E → R : f is P(FtX ) × E − measurable and   t f (s, X, x)2γ(x)Xs (dx)ds < ∞ ∀t > 0} E 0



such that V = Pm (V ) +

∞

f (s, X, x)dM (s, x).

0

Proof. Let Nt be a square integrable (FtX )-martingale under Pm . As the martingale problem (M P )m for the superprocess X is well-posed, we see from Theorem 2 and Proposition 2 of Jacod (1977) that for each n ∈ N there is a finite set of functions, N(n) N(n) X ∈ D(A), and processes, h1n , . . . , hn φ1n , . . . φn a ifinite seti of (Ft )-predictable 2 such that fn (s, X, x) = i hn (s, X)φn (x) ∈ L and t Nt = Pm (N0 ) + lim fn (s, X, x)dM (s, x) n→∞

0

in L (ΩX , FX , Pm ) for each t ≥ 0. Hence for each such t,   t 2  (s, X, x)] γ(x)Xs (dx)ds P (s, X, x) − f [f lim m n n n,n →∞ 0 t 2   t  (s, X, x)dM (s, x) = lim P (s, X, x)dM (s, x) − f f m n n  2

n,n →∞

0

0

= 0. The completeness of L2 shows that there is an f in L2 so that t t lim fn (s, X, x)dM (s, x) = f (s, X, x)dM (s, x) n→∞

0

0

in L2 (ΩX , FX , Pm ). This shows that any square integrable (FtX )-martingale under Pm is a constant plus the stochastic integral of a process in L2 with respect to the martingale measure M . This is of course equivalent to the stated result. 1,g 0

2,g 0

Corollary IV.1.3. Pm11 × Pm22 is the unique solution of (M P )m g0 . . By the uniqueness of the martingale Proof. Let P be any solution of (M P )m g0 problem for the DW-superprocess (Theorem II.5.1) we know that P(X i ∈ ·) = i,g 0

Pmii (·). If φi is a bounded measurable function on ΩiX then by the above predictable representation property ∞ i,g 0 fi (s, Xi , x)dM i(s, x) P − a.s., i = 1, 2. φi (X i ) = Pmii (φi ) + 0

(Note that the martingale measure arising in the martingale problem for Xi alone agrees with the martingale measure in (M P )m g 0 by the usual bootstrapping argument

250

Superprocesses

starting with simple functions.) The orthogonality of M 1 and M 2 implies that 1,g 0

2,g 0

P(φ1 (X1 )φ2 (X2 )) = Pm11 (φ1 )Pm22 (φ2 ). Exercise IV.1.1. Let Vti φi be the unique solution of ∂Vti γi (Vti )2 = Ai Vti − − gi0 Vti ∂t 2

V0i = φi ,

φi ∈ D(Ai ).

Let ν be a probability on MF (E1 ) × MF (E2 ) and define (LM P )νg0 in the obvious manner (ν is the law of X0 and the martingale terms are now local martingales in general). Show that any solution P of (LM P )νg0 satisfies   P exp{−Xt1 (φ1 ) − Xt2 (φ2 } = exp{−X01 (Vt1 φ1 ) − X02 (Vt2 φ2 )}dν(X0 ). Conclude that

P(Xt ∈ ·) =

1,g 0

2,g 0

Pm11 (Xt1 ∈ ·) × Pm22 (Xt2 ∈ ·)dν(m),

and then convince yourself that the appropriate version of Theorem II.5.6 shows that (LM P )νg0 is well-posed. Consider now a more general martingale problem than (M P )m g on a general ¯  = (Ω , F  , F  , P ). If mi ∈ MF (Ei ), i = 1, 2, a pair of stochastic filtered space Ω t processes (X 1 , X 2 ) ∈ Ω2 satisfies (M P )m C,D iff ∀φi ∈ D(Ai ) Xti (φi ) = mi (φi ) + Xsi (Ai φi )ds − Cti (φi ) + Dti (φi ) + Mti (φi ), where Mti (φi ) is a continuous (Ft ) − martingale such that t Xsi (γi φ2i )ds, M0i(φi ) = 0, and C i , Di are continuous, M i (φi ), M j (φj )t = δij 0

non-decreasing, adapted MF (Ei ) − valued processes, starting at 0. ¯  is as above, introduce If Ω ¯  = (Ω , F  , F  ), Ω = Ω × Ω2 , F  = F  × F 2 , Ft = (F· × F·2 )t+ , Ω t ˜ 1, X ˜ 2 ) denote points in Ω and let Π : Ω → Ω be the projection let ω  = (ω  , X map. ¯ Proposition IV.1.4. (Domination Principle) Assume X satisfies (M P )m C,D on Ω and for some θi ∈ Cb (Ei )+ , t Xri (θi 1(·))dr (as measures on Ei ) ∀s < t, i = 1, 2. (DOM ) (Dti − Dsi )(·) ≤ s

There is a probability P on (Ω , F  ) and processes (Z 1 , Z 2 ) ∈ Ω2 such that (a) If W ∈ bF  , then P(W ◦ Π|Ft ) = P (W |Ft ) ◦ Π P − a.s.

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251

¯  (b) X ◦ Π satisfies (M P )m C◦Π,D◦Π on Ω . 1 2 i (c) Z , Z are independent, Z is an (Ft ) − (Y i , γi , θi )-DW superprocess starting at mi , and Zti ≥ Xti ◦ Π on Ω ∀t ≥ 0, i = 1, 2. ¯  and W ◦ Π on Remark IV.1.5. Clearly (a) implies that W = (X, D, C, M ) on Ω ¯  have the same law. More significantly they have the same adapted distribution Ω in the sense of Hoover and Keisler (1984). This means that all random variables obtained from W , respectively W ◦ Π, by the operations of compositions with bounded continuous functions and taking conditional expectation with respect to Ft , respec¯  we may just tively Ft , have the same laws. Therefore in studying (X, C, D, M ) on Ω  ¯ as well study (X, C, D, M ) ◦ Π on Ω and hence may effectively assume (X 1 , X 2 ) is dominated by a pair of independent DW-superprocesses as above. We will do this in what follows without further ado. Sketch of Proof. The proof of Theorem 5.1 in Barlow, Evans and Perkins (1991) goes through with only minor changes. We sketch the main ideas. Step 1. DW-superprocesses with immigration. Assume i = {µ :µ is a measure on R+ × Ei , µ([0, T ] × Ei ) < ∞ ∀T > 0, µi ∈ MLF

µ({t} × Ei ) = 0 ∀t ≥ 0}, and τ ≥ 0. Consider the following martingale problem, denoted (M P )iτ,mi,µ , for a DW-superprocess with immigration µ on some (Ω, F , Ft, P): t t ∀φ ∈ D(Ai ) Xt (φ) = mi (φ) + Xs (Aθi i φ)ds + Mt (φ), φ(x)dµ(r, x) + τ

τ

t ≥ τ, where Mt (φ), t ≥ τ is a continuous (Ft )-martingale such that t Xs (γi φ2 )ds. Mτ (φ) = 0, and M (φ)t = τ

Then

(M P )iτ,mi,µ

is well-posed and the law Piτ,mi ,µ of any solution on ΩX satisfies

Piτ,mi ,µ (exp(−Xt (φ))

 = exp

t −

i mi (Vt−τ φ)

−

 i φ(x)dµ(s, x) , Vt−s

τ

where is as in Exercise IV.1.1. Moreover (ΩX , FX , FtX , Xt , Piτ,mi ,µ ) is an inhomogeneous Borel strong Markov process and (τ, mi , µ) → Piτ,mi ,µ is Borel measurable. The existence of a solution may be seen by approximating µ by a sequence of measures each supported by {t0 , . . . , tm } × Ei for some finite set of points, and taking the weak limit through an appropriate sequence of the corresponding DWsuperprocesses. For any solution to (M P )i , the formula for the Laplace functional and other properties stated above may then be derived just as in Section II.5. Note that the required measurability is clear from the Laplace functional equation, the Markov property and a monotone class argument. (Alternatively, the existence of a unique Markov process satisfying this Laplace functional equation is a special case of Theorem 1.1 of Dynkin and the corresponding martingale problem may then be derived as in Fitzsimmons (1988,1989).) Vti φ

252

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Step 2. Definition of P. Set Qiµ = Pi0,0,µ and define

t

Xsi (θ i 1(·))ds − Dti (·) + Cti (·).

Fti (·) = 0

i Then F i (ω  ) ∈ MLF P -a.s. and we can define P on Ω by 1A (ω  )Q1F 1 (ω ) (B1 )Q2F 2 (ω ) (B2 )dP(ω  ). P(A × B1 × B2 ) = Ω

˜ 1 and X ˜ 2 are independent, and X ˜ i is a This means that under P, conditional on ω  , X i  ˜ i  ˜ i. (Yi , γi , θi )-DW-superprocess with immigration Fi . Define Zt (ω , X) = Xt (ω ) + X t i ˜ For example if θi and Di are both 0, then we can think of X as keeping track of the “ghost particles” (and their descendants) killed off by C i in the X i population. When it is added to X i one should get an ordinary DW-superprocess. (a) is a simple consequence of this definition and (b) is then immediate. To prove (c) we ¯  corresponding to (M P )m and then show Z satisfies the martingale problem on Ω θ use Corollary IV.1.3. This is a straightforward calculation (see Theorem 5.1 in Barlow-Evan-Perkins(1991)). The fact that Z i dominates X i is obvious. We now state and prove a bivariate version of Dawson’s Girsanov Theorem for interactive drifts (Dawson (1978). The version given here is taken from EvansPerkins (1994). Theorem IV.1.6. Assume γi (x) > 0 for all x in Ei , i = 1, 2 and Pm1 × Pm2 -a.s., 2 t  gi (s, X, x)2 i Xs (dx)ds < ∞, ∀t > 0, γi (x) i=1 0

(IV.1.1)

so that we can define a continuous local martingale under Pm1 × Pm2 by Rtg

= exp

2  i=1

0

t

1 −gi (s, X, x) dM i,0 (s, x) − γi (x) 2

t 0

 gi (s, X, x)2 i Xs (dx)ds . γi (x)

(a) If P satisfies (M P )m g and (IV.1.1) holds P-a.s., then (IV.1.2)

 dP  g  = Rt , dP1m1 × P2m2 Ft2

and in particular there is at most one law P satisfying (M P )m g such that (IV.1.1) holds P-a.s. (b) If |gi |2 /γi (x) and |gi | are uniformly bounded for i = 1, 2 then Rtg is an (Ft2 )martingale under Pm1 × Pm2 and (IV.1.2) defines the unique law P which satisfies (M P )m g . (c) If X· ≤ X· (pointwise inequality of measures) implies

(IV.1.3) −θ γi (x) ≤ gi (t, X, x) ≤ gi (t, X  , x), i = 1, 2 for all (t, x), for some constant θ ≥ 0, then the conclusion of (b) holds.

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253

Proof. (a) Let Tn = inf{t :

2 t     gi (s, X, x)2 + 1 Xsi (ds) + 1 ds ≥ n} (≤ n). γi (x) 0 i=1

Assume P satisfies (M P )m g , (IV.1.1) holds P-a.s., and define ˜g R t∧Tn = exp

2 

t∧Tn



0

i=1

gi (s, X, x) i,gi M (ds, dx) γi (x)  1 t∧Tn gi (s, X, x)2 i − Xs (dx)ds . 2 0 γi (x)

2 ˜g Then R ·∧Tn is a uniformly integrable (Ft )-martingale under P (e.g., by Theorem ˜ g dP defines a probability on III.5.3 of Ikeda-Watanabe (1981)) and so dQn = R Tn m 2 2 (Ω , Ft ). If = denotes equality up to local martingales and φi ∈ Cb2 (Ei ), then integration by parts shows that under P, t∧Tn

 i,0 g i,gi ˜ Mt∧Tn (φ)Rt∧Tn = Mt∧Tn (φi ) − gi (s, X, x)φi(x)Xsi (dx)ds



× 1+ 0



m

t∧Tn

t∧Tn

=−





0

˜ g gi (s, X, x) M i,gi (ds, dx) R s γi (x)



˜ g gi (s, X, x)φi(x)X i (dx)ds R s s

0

+

t∧Tn

˜ g dM i,gi (φi ), M i,gi (gi /γi )s R s

0

= 0. i,0 is an (Ft2 )-local martingale. As Qn  P and quadratic Therefore under Qn , Mt∧T n variation is a path property, we also have t∧Tn i,0 j,0 M·∧Tn (φi ), M·∧Tn (φj )t = δij Xsi (γi φ2i )ds ∀t ≥ 0 Qn − a.s. 0

i,0 ˜n (φ ) is a Qn -martingale. Let Q which is uniformly bounded and hence shows Mt∧T  n i  2 2 ˜   denote the unique law on (Ω , F ) such that Qn F 2 = Qn F 2 and the conditional Tn

Tn

˜ n satisfies (M P )m and so, by law of XTn +· given FT2n is P1X 1 × P2X 2 . Then Q 0 Tn Tn 1 2 ˜ Corollary IV.1.3, Qn = Pm1 × Pm2 . Therefore (IV.1.1) implies Qn (Tn < t) = P1m1 × P2m2 (Tn < t) → 0 as n → ∞. ˜ g is an (F 2 )-local martingale under P and Since (IV.1.1) holds P-a.s., R t t

(IV.1.4)

˜ g 1(Tn ≥ t)) ˜ tg ) ≥ P(R P(R t∧Tn g ˜ ˜ g 1(Tn < t)) = P(Rt∧T ) − P(R t∧Tn n

254

Superprocesses = 1 − Qn (Tn < t) → 1 as n → ∞.

˜ g is a a P-martingale and we may define a unique law, Q, on (Ω2 , F 2 ) Therefore R t    ˜ tg dP 2 for all t > 0. Now repeat the above argument, but without by dQ F 2 = R F t

t

the Tn ’s, to see that Q = P1m1 × P2m2 . Note here that it suffices to show M i,0 (φi ) are local martingales as the proof of Corollary IV.1.3 shows the corresponding local martingale problem is well-posed. Therefore     g −1 ˜t d(P1m1 × P2m2 )F 2 = Rtg d(P1m1 × P2m2 )F 2 ∀t > 0. dPF 2 = R t

t

t

(b) Uniqueness is immediate from (a). Let Tn be as in (a), let g n (s, X, x) = 1(s ≤ Tn )(g1 (s, X, x), g2(s, X, x)), and define a probability on (Ω2 , F 2 ) by dQn = RTg n d(P1m1 × P2m2 ). Now argue just as in the proof of (a) to see that Qn solves (M P )m g n . This martingale problem shows that   t∧Tn Xsi (gi (s, X, ·))ds Qn (Xti (1)) = mi (1) + Qn 0

≤ mi (1) + cQn



t∧Tn

 Xsi (1)ds .

0

The righthand side is finite by the definition of Tn and hence so is the lefthand side. A Gronwall argument now shows that Qn (Xti (1)) ≤ mi (1)ect and therefore 2 t

   gi (s, X, x)2 + 1 Xsi (dx) + 1 ds Qn γi (x) i=1 0

≤ (c2 + 1)(m1 (1) + m2 (1))ect t + 2t ≡ K(t). This shows that Qn (Tn < t) ≤ K(t)/n → 0 as n → ∞. Argue exactly as in (IV.1.4) to see that P1m1 × P2m2 (Rtg ) = 1 and therefore Rtg is a martingale under this product measure. A simple stochastic calculus argument as in the proof of (a) shows that (IV.1.2) does define a solution of (M P )m g . Note that, as for Qn , one sees that P(Xti (1)) ≤ mi (1)ect and so M i,gi (φ) is a martingale (and not just a local martingale) because its square function is integrable. (c) Define Tn , g n and Qn as in the proof of (b). As before, Qn satisfies (M P )m gn . The upper bound on −gi allows us to apply Proposition IV.1.4 and define processes same probability space such that (X 1 , X 2 ) has law Qn and Z i ≥ X i , i = 1, 2 on the √ √ 1,θ γ 2,θ γ 1 2 (Z 1 , Z 2 ) has law Pm1 × Pm 2 . The conditions on gi show that t

t

  gi (s, X, x)2 g + (s, X, x)2 + 1 Xsi (dx)ds ≤ + 1 Xsi (dx)ds θ2 + i γi (x) γi (x) 0 0 t

 + g (s, Z, x)2 + 1 Zsi (dx)ds (IV.1.5) θ2 + i ≤ γi (x) 0 t

 gi (s, Z, x)2 + 1 Zsi (dx)ds. ≤ (θ 2 + 1) γi (x) 0

Superprocesses

255

This implies that (IV.1.6)

√ 1,θ γ1

Qn (Tn < t) ≤ Pm1

√ 2,θ γ2

× Pm 2

(Tn/(θ2 +1) < t).

√ 1,θ γ1

√ 2,θ γ × Pm2 2  P1m1 × P2m2 √ 2,θ γ Pm2 2 -a.s. and therefore the

Now (IV.1.1) and the fact that Pm1 √ 1,θ γ Pm 1 1

on Ft2 (from (b))

show that (IV.1.1) holds × expression on the righthand side of (IV.1.6) approaches 0 as n → ∞. Therefore the same is true for the lefthand side of (IV.1.6) and we can argue as in (IV.1.4) to see that Rtg is an (Ft2 )-martingale under P1m1 × P2m2 . A simple stochastic calculus argument, as in (a), shows that (IV.1.2) defines a law P which satisfies (M P )m g . Note that initially one are local martingales. As in (b) they are gets that the martingale terms in (M P )m g martingales because a simple Gronwall argument using the upper bound on −g i (use the above stopping times Tn and Fatou’s Lemma) shows that P(Xti (1)) ≤ mi (1)ect . For uniqueness assume P satisfies (M P )m g . As above we may use Proposition IV.1.4 to define processes X and Z on a common probability space such that √ √ 1,θ γ 2,θ γ X i ≤ Z i i = 1, 2, X has law P, and Z has law Pm1 1 × Pm2 2 . Recall that we saw that (IV.1.1) holds a.s. with respect to this latter law and so the calculation in (IV.1.5) shows that it holds P-a.s. as well. The uniqueness is therefore consequence of (a). Remark IV.1.7. (a) Simply take g2 = 0 in the above to get the usual univariate form of Dawson’s Girsanov theorem. (b) In Theorem 2.3 (b) of Evans-Perkins (1994) this result is stated without the monotonicity part of (IV.1.3). This is false as one can easily see by taking g1 = 1/Xs1 (1) and noting that the total mass of the solution of (M P )m g (if it existed) could now become negative because of the constant negative drift. Fortunately all the applications given there are valid because (IV.1.3) holds in each of them. m (c) If −gi ≤ c i = 1, 2 for some constant c, then (M P )m g is equivalent to (LM P )g , i,gi m i.e., (M P )g but now Mt (φi ) need only be a continuous local martingale. To see i i this, assume P satisfies (LM P )m g and let Tn = inf{t : Xt (1) ≥ n} (n > mi (1)). Then i,gi Mt∧T i (1) is a square integrable martingale because M i,gi (1)t∧Tni ≤ γi ∞ nt. We n have t

Xti (1) ≤ mi (1) + c

Xsi (1) ds + Mti,gi (1).

0

Take mean values in the above inequality at time t ∧ Tni to see that t i i E(Xt∧T (1)) ≤ m (1) + c E(Xs∧T i i (1)) ds, i n n 0

and so E(Xti (1))

i ≤ mi (1)ect . E(Xt∧T i (1)) n ≤ mi (1)ect . Therefore for each

since its square function is integrable.

By Fatou’s Lemma this implies i,gi φi ∈ D( ∆ (φi ) is an L2 -martingale 2 ), Mt

As a first application of Theorem IV.1.6 we return to

256

Superprocesses

m Example IV.1.1. Recall that (CS)ε,λ m was a special case of (M P )g with γi ≡ 1 and gi (s, X, xi) = λi pε (xi − xj )Xsj (dxj ) (j = i).

Clearly the monotonicity condition (IV.1.3) holds with θ = 0 and (IV.1.1) is clear because gi (s, X, xi) ≤ λi ε−d/2 Xsj (1). Part (c) of the above theorem therefore shows ε that the unique solution of (CS)ε,λ m is Pm , where, if Pm is the law of SBM (γ ≡ 1), then t 2

  dPεm   = exp  pε (xi − x3−i )Xs3−i (dx3−i )dM i,0 (s, xi ) − λ i d Pm1 × Pm2 Ft2 0 i=1 2  1 2 t

− λi (IV.1.7) pε (xi − x3−i )Xs3−i (dx3−i ) Xsi (dxi )ds . 2 0 (IV.1.7) defines a collection of laws {Pεm : m ∈ MF (Rd )} on (Ω2 , F 2 ). If ν is a probability on MF (Rd )2 and P satisfies (CS)ε,λ ν , that is the analogue of (CS) but with L(X0 ) = ν and Mti (φi ) now a local martingale, then one easily sees that the regular conditional probability of X given X0 satisfies (CS)ε,λ X0 for ν-a.a. X0 . Therefore this conditional law is PεX0 ν-a.s. and one can argue as in Theorem II.5.6 to see that (Ω2 , F 2 , Ft2 , Xt , Pεm ) is a Borel strong Markov process. The Borel measurability is in fact clear from (IV.1.7). Exercise IV.1.2 Assume Y is a Feller process on a locally compact separable metric space E with strongly continuous semigroup and fix γ > 0. Let Vs (ω) = ωs be the coordinate maps on ΩV = C(R+ , M1 (E)). For each V0 ∈ M( E) there is a unique ˜V ˜ V on ΩV such that under P law P 0 0 t ∀φ ∈ D(A) Vt (φ) = V0 (φ) + Vs (Aφ) ds + Mt (φ), where M (φ) is a continuous 0 t Vs (φ2 ) − Vs (φ)2 ds. (FtV )-martingale such that M0 (φ) = 0 and M (φ)t = γ 0

˜ V is the law of the Fleming-Viot process with mutation operator A (see Section P 0 10.4 of Ethier-Kurtz (1986)). For c ≥ 0 and m ∈ M1 (E) consider the following martingale problem for a law P on ΩX : t Xs (Aφ) + c(1 − Xs (1))Xs (φ) ds + Mtc (φ), where ∀φ ∈ D(A) Xt (φ) = m(φ) + 0 t Xs (φ2 ) ds. M c (φ) is an FtX -martingale such that M0c (φ) = 0 and M c (φ)t = γ 0

(a) Show there is a unique law Pc satisfying this martingale problem and find (b) Show that for any T, ε > 0, limc→∞ Pc (supt≤T |Xt (1) − 1| > ε) = 0.

dPc | X. dP0 Ft

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257

Hint. This is an exercise in one-dimensional diffusion theory–here is one approach. By a time change it suffices to show the required convergence for t √ c(1 − Ws )ds. Wt = 1 + γBt + 0

Itˆo’s Lemma implies that for any integer p ≥ 2, t t p(p − 1)γ t √ (Ws −1)p ds = p γ (Ws −1)p−1 dBs + (Ws −1)p−2 ds. (Wt −1)p +cp 2 0 0 0 Use induction and the above to conclude that for each even p ≥ 2,
t limc→∞ E( 0 (Ws − 1)p ds) = 0. Now note that the left side of the above display is a nonnegative submartingale. Take p = 4 and use a maximal inequality. Xt∧S (·) ∈ M1 (E). If φ ∈ D(A), (c) Define S = inf{t : Xt (1) ≤ 1/2} and Zt (·) = X t∧S (1)
t∧S c prove that Zt (φ) = m(φ) + 0 Zs (Aφ) ds + Nt (φ), where N c (φ) is an (FtX )martingale under Pc starting at 0 and satisfying t∧S N c (φ)t = γ (Zs (φ2 ) − Zs (φ)2 )Xs (1)−1 ds. 0

Show this implies limc→∞ Pc (Zt (φ)) = m(Pt φ). w˜ cw˜ (d) Show that Pc (Z ∈ ·)⇒P m on ΩV as c → ∞ and conclude from (b) that P ⇒Pm ˜ m as a law on ΩX because ΩV ∈ FX ). on ΩX (we may consider P Hint. Use Theorem II.4.1 to show that {Pcn (Z ∈ ·)} is tight for any cn ↑ ∞. One approach to the compact containment is as follows: Let d be a bounded metric on E ∪ {∞}, the one-point compactification of E,
1 let hp (x) = e−pd(x,∞) and gp (x) = 0 Ps hp (x) ds. Then Agp (x) = P1 hp (x) − hp (x) and (c) gives T Zs (P1 hp )ds. sup Zt (gp ) ≤ m(gp ) + sup |Ntc (gp )| + t≤T

t≤T

0

Now use the first moment result in (c) and a square function inequality to conclude that   lim lim sup Pcn sup Zt (gp ) = 0. p→∞ n→∞

t≤T

2. A Singular Competing Species Model–Dimension One √ ε ↓ 0. In this limiting regime it Consider (CS)ε,λ m as the interaction range is only the local density of the “2-population” at x that has an adverse effect on the “1-population” at x and conversely. It would seem simplest to first study this limiting model in the one-dimensional case where according to the results of Section III.4 we can expect these densities to exist. Throughout this Section Pm is the law of SBM (γ ≡ 1) and we continue to use the notation from the last Section with Ei = R, i = 1, 2. Define a Borel map U : MF (R) × R → [0, ∞] by U (µ, x) = lim sup n→∞

1 1 n µ((x − , x + ]), 2 n n

258

Superprocesses

and introduce the P × B(R)-measurable canonical densities on Ω2 , ui (t, X, x) = U (Xti , x). Then

Ωac = {X ∈ Ω2 : Xti  dx ∀t > 0, i = 1, 2} = {X ∈ Ω2 : Xti (1) = ui (t, x)dx ∀t > 0, i = 1, 2}

is a universally measurable subset of Ω2 (e.g. by Theorem III.4.4 (a) of Dellacherie and Meyer (1978)). Letting ε ↓ 0 in (CS)λ,ε m suggests the following definition: A probability P on (Ω2 , F 2 ) satisfies (CS)λm iff t  ∆φi  ds For i = 1, 2 ∀φi ∈ D(∆/2) Xti (φi ) = mi (φi ) + Xsi 2 0 t φi (x)u1 (s, x)u2 (s, x) dx ds + Mti (φi ), − λi 0

where Mti (φi ) are continuous (Ft2 ) − martingales under P such that M0i (φi ) = 0, t i j Xsi (φ2i )ds. and M (φi ), M (φj )t = δij 0 i,0

Recall that M (i = 1, 2) are the orthogonal martingale measures on Ω2 under Pm1 × Pm2 –see the notation introduced at the beginning of this Chapter. Theorem IV.2.1. Assume d = 1 and let   2 log F = (m1 , m2 ) ∈ MF (R) :

 + 1 dm1 (x1 )dm2 (x2 ) < ∞ . |x1 − x2 |

(a) For each m ∈ F , (CS)λm has a unique solution P0m given by  

dP0m  − λi  2 = exp d(Pm1 × Pm2 ) Ft 2

t

i=1

λ2 − i 2

u3−i (s, x)dM i,0(s, x) 0

t u3−i (s, x)2 ui (s, x)dxds

 .

0

In particular P0m (Ωac ) = 1. (b)(Ω2 , F 2 , Ft2 , Xt , (P0m )m∈F ) is a continuous Borel strong Markov process taking values in F. That is, for each m ∈ F , P0m (Xt ∈ F ∀t ≥ 0) = 1, m → P0m is Borel measurable, and the (Ft2 )-strong Markov property holds. w

(c) For each m ∈ F , Pεm ⇒P0m as ε ↓ 0. Proof. (a) Note first that (CS)λm is a special case of (M P )m g with gi (s, X, x) = λi u3−i (s, X, x).

Superprocesses

259

To see this, note that if P satisfies (CS)λm , then by Proposition IV.1.4 (with Di = θi = 0) we can define a process X with law P and a pair of independent superBrownian motions (γ = 1), (Z 1 , Z 2 ) on the same space so that Z i ≥ X i . As Z i  dx by Theorem III.3.8(c), the same is true of X i , and so in (CS)λm , ui (s, x)dx = Xsi (dx), and P satisfies (M P )m g as claimed. The converse implication is proved in the same way. The fact that gi can now take on the value ∞ will not alter any of the results (or proofs) in the previous section. Now check the hypotheses of Theorem IV.1.6(c) for the above choice of gi . Condition (IV.1.3) is obvious (with θ ≡ 0). For (IV.1.1), by symmetry it suffices to show that  t  (IV.2.1) Pm 1 × Pm 2 u1 (s, x)2 u2 (s, x) dx ds < ∞ ∀t > 0. 0


Recall from (III.4.1) that if mPt (x) = pt (y − x)dm(y) then under Pm1 × Pm2 , t pt−s (y − x)dM i,0 (s, y) a.s. for each t, x, ui (t, x) = mi Pt (x) + 0

where the stochastic integral is square integrable. This shows that Pmi (ui (t, xi )) = mi Pt (xi ),

(IV.2.2) and

t pt−s (y − xi )2 mi Ps (y) dy ds

Pmi (ui (t, xi )2 ) = mi Pt (xi )2 +

0 t

≤ mi Pt (xi ) + 2

(2π(t − s))−1/2 ds mi Pt (xi )

0

√ ≤ mi Pt (xi )2 + t mi Pt (xi ) ≤ mi Pt (xi )2 + mi (1).

(IV.2.3)

Now use these estimates to bound the lefthand side of (IV.2.1) by t t m1 Ps (x)2 m2 Ps (x) dx ds + m1 (1)m2 Ps (x) dx ds. 0

0

The second term is m1 (1)m2 (1)t and so is clearly finite for all t > 0 for any pair of finite measures m. Bound m1 Ps (x)2 by m1 (1)s−1/2 m1 Ps (x) and use ChapmanKolmogorov to see that the first term is at most t m1 (1) s−1/2 p2s (y1 − y2 )m1 (dy1 )m2 (dy2 ) ds 0    + 4t ≤ m1 (1) dm1 (y1 )dm2 (y2 ) 1 + log |y1 − y2 |2 < ∞ if m ∈ F. (b) Let Z be the pair of independent dominating SBM’s constructed in (a). Since Zti has a continuous density on compact support for all t > 0 a.s. (Theorem III.4.2(a)

260

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and Corollary III.1.4), clearly Zt ∈ F for all t > 0 a.s. and hence the same is true for X P0m a.s. The Borel measurability in m is clear from the Radon-Nikodym derivative provided in (a) and the strong Markov property is then a standard consequence of uniqueness (see, e.g. the corresponding discussion for Pεm at the end of the last section). (c) Write Pm for Pm1 × Pm2 . Let Rtε be the Radon-Nikodym derivative in (IV.1.7) and Rt0 be that in (a) above. It suffices to show Rtε → Rt0 in Pm -probability because as these non-negative random variables all have mean 1, this would imply L1 convergence. To show this convergence, by symmetry it clearly suffices to prove t [Xs2 Pε (x) − u2 (s, x)]2 u1 (s, x) dx ds → 0 in Pm -probability as ε ↓ 0. 0

If δ > 0 is fixed, the fact that (Xs2 , s ≥ δ) has a jointly continuous uniformly bounded density shows that t [Xs2 Pε (x) − u2 (s, x)]2 u1 (s, x) dx ds → 0 in Pm -probability as ε ↓ 0. δ

Therefore it suffices to show  δ  lim sup Pm [Xs2 Pε (x)2 + u2 (s, x)2 ]u1 (s, x) dx ds = 0. δ↓0 0<ε<1

0

The argument in (a) easily handles the u2 (s, x)2 term, so we focus on the Xs2 Pε (x)2 term. Use (IV.2.2) and (IV.2.3) to see that  δ  Xs2 Pε (x)2 u1 (s, x) dx ds Pm 0



δ

pε (y − x)m2 Ps (y)2 dy m1 Ps (x) dx ds + m2 (1)m1 (1)δ

≤ 0

≤

δ

m2 (1)s−1/2

m2 Ps+ε (x)m1 Ps (x) dx ds + m1 (1)m2 (1)δ

0

→ 0 as δ ↓ 0, by the same argument as that at the end of the proof of (a). Remark IV.2.2. In (CS)λm we may restrict the test functions φi to Cb∞ (R). To see this, first recall from Examples II.2.4 that this class is a core for D(∆/2). Now suppose the conclusion of (CS)λm has been verified for Cb∞ (R) and for a sequence bp

∆ n of functions {(φn1 , φn2 )} in D(∆/2)2 such that (φni , ∆ 2 φi )→(φi , 2 φi ) as n → ∞ for λ i i = 1, 2. It follows from (CS)m for φi ≡ 1 that E(Xs (1)) ≤ mi (1). Therefore by Dominated Convergence   T

E(sup(Mti (φni ) − Mti (φi ))2 ) ≤ cE t≤T

Xsi ((φni − φi )2 ) ds

→ 0 as n → ∞.

0

By Dominated Convergence it is now easy to take limits in (CS)λm to see that this conclusion persists for the limiting functions (φ1 , φ2 ). This establishes the claim.

Superprocesses

261

We also showed in the proof of (a) above that (CS)λm is equivalent to (M P )m g with gi = λi u3−i ≥ 0. Hence by Remark IV.1.7(c), (CS)λm remains unchanged if we only assumed that M·i (φi ) are continuous (Ft2 )-local martingales. One can easily reformulate (CS)λm as a stochastic pde. Assume W1 , W2 are ¯ = (Ω, F , Ft, P). Recall that CK (R) is the space of independent white noises on Ω continuous functions on R with compact support equipped with the sup norm. A pair of non-negative processes {u1 (t, x), u2(t, x) : t > 0, x ∈ R}, is a solution of √ ˙ ∂ui ∆ui = − λi u1 u2 + ui W i , ui (0+, x)dx = mi . ∂t 2

(SPDE)λm iff for i = 1, 2,

(i) {ui (t, ·) : t > 0} is continuous and (Ft ) − adapted with values in CK (R). t φ  − λi u1 (s)u2 (s), φ ds ui (s), (ii)ui (t), φ ≡ ui (t, x)φ(x)dx = mi (φ) + 2 0 t

φ(x) ui (s, x)dWi (s, x), ∀t > 0 a.s. ∀φ ∈ Cb2 (R). + 0

As in Remark III.4.1 this implies a.s. Xti (dx) ≡ ui (t, x)dx−→mi as t ↓ 0.

(IV.2.4)

Proposition IV.2.3. Assume m ∈ F . (a) If (u1 , u2 ) satisfies (SPDE)λm , and X is given by (IV.2.4), then L(X) = P0m . In particular, the law of u on C((0, ∞), CK (R)2 ) is unique. ¯  = (Ω , F  , F  , P ) such that if (b) There is an Ω t   2 ¯ Ω = (Ω × Ω , F 2 × F  , F·2 × F· t+ , P0m × P ) and Π : Ω2 × Ω → Ω is the projection map, then there is a pair of independent ˙ 2 on Ω ¯ such that (u1 , u2 ) ◦ Π solves (SPDE)λ on Ω. ¯ ˙ 1, W white noises, W m Proof. (a) The weak continuity of X follows from (IV.2.4) as in the proof of Theorem III.4.2(c). It now follows from Remark IV.2.2 that X satisfies (CS)λm and hence has law P0m by Theorem IV.2.1. The second assertion now follows as in the univariate case (Corollary III.4.3(c)). (b) Let un,i (t, X, x) = n2 Xti ((x − n1 , x + n1 ]). We know P0m  Pm1 × Pm2 and under the latter measure ui (t, x) is the jointly continuous density of Xti on (0, ∞) × R (Theorem III.4.2(a)), and {(t, x) : ui (t, x) > 0, t ≥ δ} is bounded for every δ > 0 (Corollary III.1.7). It follows that Pm1 × Pm2 -a.s. and therefore P0m -a.s. for every δ > 0, and i = 1, 2, sup

sup |un,i (t, x) − un ,i (t, x)|

x∈R t∈[δ,δ −1 ]

= sup

sup

x∈Q t∈Q∩[δ,δ −1 ]

|un,i (t, x) − un ,i (t, x)| → 0 as n, n → ∞,

262

Superprocesses

and ∃R such that supt∈[δ,δ−1 ] Xti (B(0, R)c) = 0. It follows that (i) holds P0m -a.s. It remains to show that (ii) holds on this larger space. Choose Ω carrying two ¯ by independent white noises, W1 , W2 on R+ × R. Define Wi on Ω t 1(ui (s, X, x) > 0) Wi (ω  , X)t(A) = dM i (X)(s, x) 1A (x) ui (s, X, x) 0 t + 1A (x)1(ui (s, X, x) = 0)dWi (ω  )(s, x). 0

¯ and As in Theorem III.4.2(b), (W1 , W2 ) are independent white noises on Ω 1 2 λ ¯ (u , u ) ◦ Π satisfies (SPDE)m on Ω. Note the independence follows from the orthogonality of the martingales W1 (t)(A) and W2 (t)(B) for each A and B because these are Gaussian processes in (t, A). Here is a univariate version of the above result which may be proved in the same manner. If σ 2 , γ > 0, λ ≥ 0, and θ ∈ R, consider (SP DE)

σ 2 ∆u √ ∂u ˙ + θu − λu2 , = + γuW ∂t 2

u0+ (x)ds = m(dx),

where m ∈ MF (R), and the above equation is interpreted as before. (x), to denote the canonical density In the next result we also use ut (x) = dX dx of the absolutely continuous part of Xt on the canonical space of paths ΩX (defined as before). Proposition IV.2.4. Assume  + 1 dm(x1 )dm(x2 ) < ∞. log |x1 − x2 | (a) There is a filtered space (Ω, F , Ft, P) carrying a solution of (SPDE). (b) If u is any solution of (SPDE) and P is the law of t → ut (x)dx on ΩX , then  t  1 t dP  (θ − λu(s, x))dM (s, x) − (θ − λu(s, x))2 Xs (dx)dx .  X = exp dPm Ft 2 0 0 Here Pm is the law of super-Brownian motion starting at m with spatial variance σ 2 , 0 drift and branching rate γ, and dM (s, x) is the associated martingale measure. In particular the law of u on C((0, ∞), CK (R)2 ) is unique. The above result was pointed out by Don Dawson in response to a query of Rick Durrett. Durrett’s question was prompted by his conjecture that the above SPDE arises in the scaling limit of a contact process in one dimension. The conjecture was confirmed by Mueller and Tribe (1994). 3. Collision Local Time To study (CS)λm in higher dimensions we require an analogue of u1 (s, x)u2 (s, x)dsdx which will exist in higher dimensions when the measures in question will not have densities. This is the collision local time of a pair of measurevalued processes which we now define.

Superprocesses

263

Definition. Let X = (X 1 , X 2 ) be a pair of continuous MF (Rd )-valued processes on a common probability space and let pt denote the standard Brownian transition density. The collision local time (COLT) of X is a continuous non-decreasing MF (Rd )-valued process Lt (X) such that for any φ ∈ Cb (Rd ) and t ≥ 0, t  x1 + x 2 P φ Lεt (X)(φ) ≡ pε (x1 − x2 )Xs1 (dx1 )Xs2 (dx2 )ds→Lt (X)(φ) 2 0 as ε → 0. Definition. The graph of an MF (Rd )-valued process (Xt , t ≥ 0) is G(X) = ∪δ>0 cl{(t, x) : t ≥ δ, x ∈ S(Xt )} ≡ ∪δ>0 Gδ (X) ⊂ R+ × Rd . Remarks IV.3.1. (a) Clearly the process L(X) is uniquely defined up to null sets. It is easy to check that L(X)(ds, dx) is supported by G(X 1 ) ∩ G(X 2 ). This random measure gauges the intensity of the space-time collisions between the populations X 1 and X 2 and so can be used as a means of introducing local interactions between these populations. See the next section and Dawson et al (2000a) for examples. (b) If Xsi (dx) = ui (s, x)dx, where ui is a.s. bounded on [0, t] × Rd , then an easy application of Dominated Convergence, shows that  t  u1 (s, x)u2 (s, x)ds dx. Lt (X)(dx) = 0

However Lt (X) may exist even for singular measures as we will see in Theorem IV.3.2 below. (c) The definition of collision local time remains unchanged if Lεt (X)(φi ) is replaced
t
pε (x1 − x2 )Xsj (dxj )φi (xi )Xsi (dxi )ds (i = j). This is easy with Lε,i t (X)(φi ) = 0 to see by the uniform continuity of φ on compact sets. Throughout this Section we will assume Z i is an (Ft ) − (SBM )(γi) starting at mi ∈ MF (Rd ), i = 1, 2, defined on (Ω, F , Ft, P), and (Z 1 , Z 2 ) are independent.   r −β if β > 0 1 2 Let Zt = Zt × Zt . Recall from Section III.5 that gβ (r) = 1 + log+ 1r if β = 0 .  1 if β < 0

(H1 )

Theorem IV.3.2. (a) If d ≤ 5, mi = 0, and (IC) gd−2 (|z1 − z2 |)m1 (dz1 )m2 (dz2 ) < ∞ gd−1 (|z1 − z2 |)m1 (dz1 )m2 (dz2 ) < ∞

if d ≤ 4 if d = 5,

then Lt (Z) exists, is not identically 0 and satisfies lim  sup |Lεt (Z)(φ) − Lt (Z)(φ)| 2 = 0 ∀T > 0, φ ∈ bB(Rd ). ε↓0

t≤T

264

Superprocesses

In particular, P(G(Z 1 ) ∩ G(Z 2 ) = ∅) > 0. (b) If d ≥ 6, then G(Z 1 ) ∩ G(Z 2 ) = ∅ a.s. We will prove this below except for the critical 6-dimensional case whose proof will only be sketched. Lemma IV.3.3. If d ≥ 2, there is a constant C = C(d, γ1 , γ2 ), and for each δ > 0, a random r1 (δ, ω) > 0 a.s. such that for all 0 ≤ r ≤ r1 (δ),  1 2+2/d . 1(|z1 − z2 | ≤ r)Zt1 (dz1 )Zt2 (dz2 ) ≤ C(sup Zt1 (1) + 1)r 4−4/d log sup r t t≥δ Proof. We defer this to the end of this Section. It is a nice exercise using the results of Chapter III but the methods are not central to this Section. Clearly if d = 1 the above supremuim is a random multiple of r by Theorem III.4.2. Corollary IV.3.4. If d ≥ 2 and 0 ≤ β < 4 − 4/d, then with probability 1, lim sup gβ (|z1 − z2 |)1(|z1 − z2 | ≤ ε)Zt1 (dz1 )Zt2 (dz2 ) = 0 ∀δ > 0 ε↓0 t≥δ



and t →

gβ (|z1 − z2 |)Zt1 (dz1 )Zt2 (dz2 ) is continuous on (0, ∞).

In particular, this is the case for β = d − 2 and d ≤ 5. Proof. Define a random measure on [0, ∞) by Dt (A) = Zt ({(z1 , z2 ) : |z1 − z2 | ∈ A}). If 0 < β < 4 − 4/d and ε < r(δ, ω), then an integration by parts and Lemma IV.3.3 give gβ (|z1 − z2 |)1(|z1 − z2 | ≤ ε)Zt1 (dz1 )Zt2 (dz2 ) sup t≥δ



= sup gβ (r)Dt ([0, r])|ε0+ + β t≥δ

≤

C(sup Zt1 (1) t

ε

r −1−β Dt ([0, r]) dr



0

ε

 1 1 + 1) ε−β+4−4/d (log+ )2+2/d + β r 3−β−4/d (log+ )2+2/d dr ε r 0

→ 0 as ε ↓ 0, by our choice of β. It follows that for all 0 ≤ β < 4 − 4/d, lim sup (gβ (|z1 − z2 |) − gβ (|z1 − z2 |) ∧ M ) dZt1 dZt2 = 0 a.s. M →∞ t≥δ

(if β
= 0 we simply compare with a β > 0). The weak continuity of Zt shows that t → gβ (|z1 − z2 |) ∧ M dZt is a.s. continuous and the second result follows. Throughout the rest of this Section we assume d (H2 ) X = (X 1 , X 2 ) satisfies (M P )m C,0 for some C with Ei = R and Ai = ∆/2 ¯ = (Ω, F , Ft, P). on Ω

Superprocesses

265

Apply Proposition IV.1.4 with Di = 0 to see that by enlarging the space we may ¯ such assume there is a pair of independent SBM’s (Z 1 , Z 2 ) as in (H1 ) defined on Ω i i 1 2 that Xt ≤ Zt for all t ≥ 0 and i = 1, 2. Set Xt = Xt × Xt . We first derive a martingale problem for X and then construct Lt (X) by means of a Tanaka formula. 

Notation. ∆ 2 is the generator of the standard 2d-dimensional Brownian motion  and Pt is its semigroup.  Lemma IV.3.5. For any φ ∈ D(∆/2), t φ(x1 , x2 )[Xs1 (dx1 )M 2 (ds, dx2 ) + Xs2 (dx2 )M 1 (ds, dx1 )] Xt (φ) =X0 (φ) + 0 t (IV.3.1) φ(x1 , x2 )[Xs1 (dx1 )C 2 (ds, dx2 ) + Xs2 (dx2 )C 1 (ds, dx1 )] − 0



t

Xs (

+ 0

 ∆φ ) ds. 2

 Proof. Step 1. φ(x1 , x2 ) = φ1 (x1 )φ2 (x2 ), φi ∈ D(∆/2). Then Xt (φ) = Xt1 (φ1 )Xt2 (φ2 ) and the result follows from (M P )m C,0 by an integration by parts.  Step 2. φ(x) = Pε ψ(x), where ψ ∈ D(∆/2). Then there is a sequence of finite Riemann sums of the form  pε (y1i,n − x1 )pε (y2i,n − x2 )ψ(y1i,n , y2i,n )∆n , φn (x1 , x2 ) = y1i,n ,y2i,n bp

such that φn →φ and  ∆   ∆ φn (x1 , x2 ) = (pε (y1i,n − ·)pε (y2i,n − ·))(x1 , x2 )ψ(y1i,n, y2i,n )∆n 2 2 i,n i,n y1 ,y2

bp

→

 ∆ Pε ψ(x1 , x2 ). 2

By Step 1, (IV.3.1) holds for each φn . Now let n → ∞ and use Dominated Convergence to obtain this result for φ.  Step 3. φ ∈ D(∆/2).   bp bp     P φ = Pεn ∆ φ →∆ φ as n → ∞. Now Let εn ↓ 0 and note that Pεn φ→φ and ∆ 2 εn 2 2 use (IV.3.1) for Pε φ (from Step 2) and let n → ∞ to derive it for φ. n

 Let φ ∈ CK (Rd ) and apply the above result to φε ∈ D(∆/2), which is chosen so that t   ∆ φε ds = −Lεt (φ). Xs (IV.3.2) 2 0

266

Superprocesses

This will be the case if  ∆ x1 − x 2 φε (x1 , x2 ) = −pε (x1 − x2 )φ( ) ≡ ψε (x1 , x2 ). 2 2

(IV.3.3)

 λ denote the 2d-dimensional Brownian resolvent for λ ≥ 0 and assume d > 2. Let U  0 ψε (x) ∈ D(∆/2)  satisfies (IV.3.3). If Bs = (Bs1 , Bs2 ) By Exercise II.2.2, φε (x) = U 1 2 1 2 is a 2d-dimensional Brownian motion, then B √+B and B √−B are independent d2 2 dimensional Brownian motions and so a simple calculation yields  ∞  B1 + B2   s ds pε (Bs1 − Bs2 )φ s φε (x1 , x2 ) = E x1 ,x2 2 0 ∞ x − x  x1 + x2  1 2 = 21−d pε/4+u (IV.3.4) Pu φ( du 2 2 0 ≡ Gε φ(x1 , x2 ). We may use (IV.3.2) in Lemma IV.3.5 and conclude that Xt (Gε φ) = X0 (Gε φ) t + Gε φ(x1 , x2 )[Xs1 (dx1 )M 2 (ds, dx2 ) + Xs2 (dx2 )M 1 (ds, dx1 )] 0 t (T )ε Gε φ(x1 , x2 )[Xs1 (dx1 )C 2 (ds, dx2 ) + Xs2 (dx2 )C 1 (ds, dx1 )] − 0

− Lεt (φ)

∀t > 0,

for d > 2. bp

(IV.3.4) shows that Gε φ is defined for any φ ∈ bB(Rd ) and that φn →φ implies bp

Gε φn →Gε φ. Now use Dominated Convergence to extend (T )ε to all φ ∈ bB(Rd ). A similar argument with ∞ x + x  x − x  1 2 1 2  λ ψε (x1 , x2 ) = 21−d e−2λu pε/4+u Pu φ du Gλ,ε φ(x1 , x2 ) ≡ U 2 2 0 in place of Gε φ = G0,ε φ shows that for any φ ∈ bB(Rd ), Xt (Gλ,ε φ) = X0 (Gλ,ε φ) t + Gλ,ε φ(x1 , x2 )[Xs1 (dx1 )M 2 (ds, dx2 ) + Xs2 (dx2 )M 1 (ds, dx1 )] 0 t − (T )λ,ε Gλ,ε φ(x1 , x2 )[Xs1 (dx1 )C 2 (ds, dx2 ) + Xs2 (dx2 )C 1 (ds, dx1 )] 0 t +λ Xs (Gλ,ε φ) ds − Lεt (φ) ∀t > 0, for d ≥ 1. 0

As we want to let ε ↓ 0 in the above formulae, introduce ∞ x + x  x − x  1 2 1 2 e−2λu pu Pu φ du, Gλ,0 φ(x1 , x2 ) = 21−d 2 2 0

G0 φ = G0,0 φ,

Superprocesses

267

when this integral is well-defined, as is the case if φ ≥ 0. A simple integration shows that for any ε ≥ 0, (IV.3.5)

Gε |φ|(x1 , x2 ) ≤ φ∞ G0 1(x1 , x2 ) = φ∞ kd gd−2 (|x1 − x2 |) if d > 2,

where kd = Γ(d/2 − 1)2−1−d/2 π −d/2 . Therefore G0 φ(x1 , x2 ) is finite when φ is bounded, x1 =  x2 , and d > 2. Lemma IV.3.6. Let φ ∈ bB(Rd ) and d > 2. Then |Gε φ(x1 , x2 ) − G0 φ(x1 , x2 )| ≤ φ∞ cd min(|x1 − x2 |2−d , ε|x1 − x2 |−d ). If φ ≥ 0, limε↓0 Gε φ(x) = G0 φ(x) (≤ ∞) for all x and G0 φ is lower semicontinuous. Proof. |Gε φ(x1 , x2 ) − G0 φ(x1 , x2 )| ∞ |pε/4+u ((x1 − x2 )/2) − pu ((x1 − x2 )/2)| du ≤ φ∞ 0



∞

≤ φ∞ ≤ φ∞

0





u

∞

   ∂pv ((x1 − x2 )/2) dv du  ∂v

u+ε/4 

ε pv ((x1 − x2 )/2)[(x1 − x2 )2 v −2 /8 + d(2v)−1 ] dv 4 ε/4  pv ((x1 − x2 )/2)[(x1 − x2 )2 (8v)−1 + d/2]dv +

ε/4

0

≤ φ∞ cd ε

(x1 −x2 )2 /2ε

0

e−y y d/2−1 (y + d) dy|x1 − x2 |−d ∞  + e−y y d/2−2 [y + (d/2)]dy|x1 − x2 |2−d , (x1 −x2 )2 /2ε

where we substituted y = (x1 − x2 )2 (8v)−1 in the last line . The integrand in the first term of the last line is both bounded and integrable and so the first term is at most cd εφ∞ |x1 − x2 |−d min((x1 − x2 )2 (2ε)−1 , 1). The integrand in the second term is at most c(y −2 ∧ 1) and so the second term is bounded by cd min(|x1 − x2 |2−d , ε|x1 − x2 |−d ). This gives the first inequality and so for the second result we need only consider x = (x1 , x1 ). This is now a simple consequence of Monotone Convergence. The lower semicontinuity of G0 φ follows from the fact that it is the increasing pointwise limit of the sequence of continuous functions ∞ pu ((x1 − x2 )/2)Pu φ((x1 + x2 )/2) du. 2−n

268

Superprocesses

Lemma IV.3.7. If 3 ≤ d ≤ 5, then for each t > 0 there is a cd (t) so that  t 2  E gd−2 (|z1 − z2 |)Zs1 (dz1 ) Zs2 (dz2 )ds 0 [g2(d−3) (|z1 − z2 |) + 1] dm1 (z1 )dm2 (z2 ). ≤ cd (t)
Proof. We may assume t ≥ 1. Recall that m2 Ps (x) = ps (y − x)m2 (dy). Use the first and second moment calculations in Exercise II.5.2 to see that the above expectation is t 2 gd−2 (|z1 − z2 |)m1 Ps (z1 )dz1 m2 Ps (z2 ) dz2 ds 0 t s  Ps−u (gd−2 (| · −z2 |))(z1 )2 m1 Pu (z1 )dz1 du m2 Ps (z2 ) dz2 ds + 0

0

(IV.3.6) ≡ I1 + I2 . Use



∞

(IV.3.7)

pu (x) du = k(d)gd−2 (|x|) 0

and Chapman-Kolmogorov to see that t ∞ ∞  m1 Ps+u (z2 )m1 Ps+u (z2 )du du m2 Ps (z2 )dz2 ds I1 = cd 0 0 u t ∞ ∞  (s + u )−d/2 du m1 (1) p2s+u (z1 − z2 )dm1 (z1 )dm2 (z2 )duds ≤ cd 0 0 u t ∞  (v − s)1−d/2 pv (z1 − z2 )dvds dm1 (z1 )dm2 (z2 ) ≤ cd m1 (1) 0 ∞ 2s  v 1−d/2 (v ∧ t)pv (z1 − z2 )dv dm1 (z1 )dm2 (z2 ) ≤ cd m1 (1) 0 t ∞  v 2−d/2 pv (z1 − z2 )dv + t v 1−d dv dm1 (z1 )dm2 (z2 ). ≤ cd m1 (1) 0

t

A routine calculation now shows that (recall t ≥ 1 to handle the second term) 
(|z1 −z2 |6−2d+ 1) dm1 (z1 )dm2 (z2 ) if d > 3 (IV.3.8) I1 ≤ cd m1 (1)
+ 2t (log |z1 −z2 | + 1) dm1 (z1 )dm2 (z2 ) if d = 3. For I2 , note first that (IV.3.7) implies Ps−u (gd−2 (| · −z2 |))(z1 ) = k(d)

∞

s−u

pv (z1 − z2 ) dv,

Superprocesses and so





t

I2 = 2k(d)2 0



ds 0

∞

du 0

∞

dv s−u



s



∞

du 0



t

≤ cd



s

ds

dv 

v

dv v 1−d/2



269



pv (z1 − z2 )pv  (z1 − z2 )  m1 Pu (z1 )m2 Ps (z2 )dz1 dz2

 pu+v+s (z1 − z2 )m1 (dz1 )m2 (dz2 ) .

s−u

Use the fact that pu+v+s (x) ≤ 2d/2 p2(u+v) (x) for s ≤ u + v and integrate out s ∈ [u, (u + v) ∧ t] in the above to get t ∞  I2 ≤ cd v 1−d/2 min(v, t − u)p2(u+v) (z1 − z2 ) dvdu m1 (dz1 )m2 (dz2 ) 0 0 ∞ w∧t  (w − u)1−d/2 ((w ∧ t) − u) du p2w (z1 − z2 ) dw m1 (dz1 )m2 (dz2 ) ≤ cd 0 0 ∞  (IV.3.9) ≤ cd (w ∧ t)3−d/2 p2w (z1 − z2 ) dw dm1 (z1 )dm2 (z2 ). 0

A change of variables now gives (recall t ≥ 1) ∞ ∞ ∞ 

(w ∧ t)3−d/2 p2w (∆) dw ≤ cd ∆8−2d xd−5 e−x dx + t3−d/2 w−d/2 dw 0 ∆2 /4t t  −2 if d = 5  ∆ + 1  ≤ cd log+ ∆4t2 + 1 if d = 4  t if d = 3. Use this in (IV.3.9) to see that 

−2 if d = 5  

(|z1 −z2 | +1) m1 (dz1 )m2 (dz2 ) + 1 I2 ≤ cd (t) (log |z1 −z2 | + 1) m1 (dz1 )m2 (dz2 ) if d = 4   m1 (1)m2 (1) if d = 3. Combine this with (IV.3.8) and (IV.3.6) to complete the proof. Theorem IV.3.8. Assume X satisfies (H2 ) where d ≤ 5 and m1 , m2 satisfy (IC). (a) Lt (X) exists and for any φ ∈ bB(Rd ), (IV.3.10)

L2

sup |Lεt (X)(φ) − Lt (X)(φ)|→0 as ε ↓ 0

for all T > 0.

t≤T

(b) If λ = 0 and d ≥ 3, or λ > 0 and d ≥ 1, then for any φ ∈ bB(Rd ),

270

Superprocesses t Gλ,0 φ(x1 , x2 )[Xs1 (dx1 )M 2 (ds, dx2 )

Xt (Gλ,0 φ) = X0 (Gλ,0 φ) + 0

+ Xs2 (dx2 )M 1 (ds, dx1 )]

t −

(T )

Gλ,0 φ(x1 , x2 )[Xs1 (dx1 )C 2 (ds, dx2 ) 0

+ Xs2 (dx2 )C 1 (ds, dx1 )]



t

Xs (Gλ,0 φ) ds − Lt (X)(φ) ∀t ≥ 0

+λ

a.s.

0

Each of the above processes are a.s. continuous in t ≥ 0. The second term on the right-hand side is an L2 (Ft )-martingale and each of the other processes on the right-hand side has square integrable total variation on bounded time intervals. Proof. We give the proof for λ = 0 and d ≥ 3. The extra term involved when λ > 0 is very easy to handle and for d ≤ 3 the entire proof simplifies considerably by means of a systematic use of Theorem III.3.4 (the reader may want to consider only this case, especially as the interactive models discussed in the next Section will only exist in these dimensions). Let φ ∈ bB(Rd )+ and note that it suffices to prove the Theorem for such a non-negative φ. Consider the limit of each of the terms in (T )ε as ε ↓ 0. (IC) and (IV.3.5) allow us to use Dominated Convergence and conclude from Lemma IV.3.6 that G0 φdm1 dm2 . Gε φdm1 dm2 = (IV.3.11) lim ε↓0

Let

t

Ntε (φ) =

Gε φ(x1 , x2 )[Xs1 (dx1 )M 2 (ds, dx2 )+Xs2 (dx2 )M 1 (ds, dx1 )],

ε, t ≥ 0.

0

Note that Lemma IV.3.7, (IV.3.5) and the domination X i ≤ Z i show that Ntε (φ) is a well-defined continuous square-integrable martingale even for ε = 0. Similarly, Lemmas IV.3.6 and IV.3.7, this domination, and Dominated Convergence show that for any T > 0, E(sup(Ntε (φ) − Nt0 (φ))2 ) t≤T

(IV.3.12)

 ≤ cE



T

γ2

0

+



T

γ1 0

→ 0 as ε ↓ 0.

|Gε φ − G0 φ|(x1 , x2 )Zs1 (dx1 )

2

|Gε φ − G0 φ|(x1 , x2 )Zs2 (dx2 )

Zs2 (dx2 )ds

2

Zs1 (dx1 )ds



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271

If C(ds, dx1 , dx2 ) = Xs1 (dx1 )C 2 (ds, dx2 ) + Xs2 (dx2 )C 1 (ds, dx1 ) then (T )ε implies for any t > 0, t Gε 1(x1 , x2 )C(ds, dx1 , dx2 ) ≤ m1 × m2 (Gε 1) + Ntε (1) 0 L2

→m1 × m2 (G0 1) + Nt0 (1), the last by (IV.3.11) and (IV.3.12). Fatou’s lemma and the equality in (IV.3.5) now show that  t 2  (IV.3.13) E gd−2 (|x1 − x2 |)C(ds, dx1 , dx2 ) < ∞ ∀t > 0. 0

This allows us to apply Lemma IV.3.6 and Dominated Convergence to conclude  t  |Gε φ − G0 φ(x1 , x2 )|C(ds, dx1 , dx2 )2 = 0. (IV.3.14) lim E ε↓0

0

(T )ε shows that Xt (Gε φ) ≤ X0 (Gε φ) + Ntε (φ) for all t ≥ 0 a.s. Let ε ↓ 0, use Lemma IV.3.6 and Fatou’s Lemma on the left-hand side, and (IV.3.11) and (IV.3.12) on the right-hand side to see that Xt (G0 φ) ≤ X0 (G0 φ) + Nt0 (φ) < ∞ ∀t ≥ 0 a.s.

(IV.3.15)

Take φ = 1 in the above inequality, recall that Nt0 (1) is an L2 -martingale, and use the equality in (IV.3.5) to get 2   gd−2 (|x1 − x2 |)Xt1 (dx1 )Xt2 (dx2 ) < ∞ ∀T > 0. (IV.3.16) E sup t≤T

The bound in Lemma IV.3.6 shows that for any T, δ, η > 0, if Sδ = {(x1 , x2 ) ∈ R2d : |x1 − x2 | ≤ δ}, then sup Xt (|Gε φ − G0 φ|) t≤T

≤ sup Xt (|Gε φ − G0 φ|1Sδc ) t≤T (IV.3.17) + cd φ∞ sup gd−2 (|x1 − x2 |)1Sδ (x1 , x2 )Xt (dx1 , dx2 ) t≤T

−d gd−2 (|x1 − x2 |)1Sδ (x1 , x2 )Xt (dx1 , dx2 ) ≤ cd φ εδ sup Xt (1) + sup t≤T η≤t≤T  + sup gd−2 (|x1 − x2 |)1Sδ (x1 , x2 )Xt (dx1 , dx2 ) . t<η


Write Xt (gd−2 ) for gd−2 (|x1 − x2 |)Xt (dx1 , dx2 ). The lower semicontinuity of (x1 , x2 ) → gd−2 (|x1 − x2 |) (take φ = 1 in Lemma IV.3.6 and use the equality in (IV.3.5)) and the weak continuity of X show that lim inf t↓0 Xt (gd−2 ) ≥ X0 (gd−2 ).

272

Superprocesses

On the other hand (IV.3.15) with φ = 1 implies lim supt↓0 Xt (gd−2 ) ≤ X0 (gd−2 ) a.s., and so lim Xt (gd−2 ) = X0 (gd−2 ) a.s.

(IV.3.18)

t↓0

Choose δn ↓ 0 so that X0 ({(x1 , x2 ) : |x1 − x2 | = δn }) = 0. Weak continuity then implies limt↓0 Xt (gd−2 1Sδc ) = X0 (gd−2 1Sδc ) and so (IV.3.18) gives n

(IV.3.19)

n

lim Xt (gd−2 1Sδn ) = X0 (gd−2 1Sδn ) a.s. t↓0

Let ε0 > 0 and first choose an natural number N0 so that the right-hand side is at most ε0 for n ≥ N0 . Next use (IV.3.19) to choose η = η(ε0 ) so that (IV.3.20)

∀n ≥ N0

sup Xt (gd−2 1Sδn ) ≤ sup Xt (gd−2 1SδN ) < 2ε0 . t<η

0

t<η

By Corollary IV.3.4 we may omit a P-null set and then choose N1 (η) ≥ N0 so that sup Xt (gd−2 1SδN ) < ε0 .

(IV.3.21)

1

t≥η

Now take δ = δN1 and η = η(ε0 ) in (IV.3.17). By (IV.3.20) and (IV.3.21) we see that outside a null set for ε < ε(ε0 ), the right-hand side of (IV.3.17) will be at most φ∞ cd 4ε0 . We have proved (IV.3.22)

lim sup Xt (|Gε φ − G0 φ|) = 0 ∀T > 0 a.s. and in L2 , ε↓0 t≤T

where Dominated Convergence, Lemma IV.3.6, and (IV.3.16) are used for the L2 convergence. (IV.3.11), (IV.3.12), (IV.3.14) and (IV.3.22) show that each term in (T )ε , except perhaps for Lεt (φ), converges uniformly in compact time intervals in L2 . Therefore ˜ t (φ) : t ≥ 0}, so that there is an a.s. continuous process {L (IV.3.23)

˜ t (φ)|2 = 0 lim  sup |Lεt (φ) − L ε↓0

t≤T

∀T > 0.

Take L2 limits uniformly in t ≤ T in (T )ε to see that (IV.3.24) t ˜ t (φ) G0 φ(x1 , x2 )C(ds, dx1 , dx2 )−L Xt (G0 φ) = m1 ×m2 (φ)+Nt0 (φ)−

∀t ≥ 0 a.s.,

0

where each term is a.s. continuous in t, Nt0 (φ) is an L2 martingale and the last two terms have square integrable total variation on compact time intervals. To complete the proof we need to show there is a continuous increasing MF (Rd )valued process Lt (X) such that (IV.3.25)

˜ t (φ) ∀t ≥ 0 a.s. for all φ ∈ bB(Rd )+ . Lt (X)(φ) = L

Note that (IV.3.23) then identifies L(X) as the collision local time of X as the notation suggests. Let D0 be a countable dense set in C (Rd ) = {φ ∈ Cb (Rd ) : φ has a limit at ∞}

Superprocesses

273

containing 1. Choose εn ↓ 0 and ω outside a null set so that (IV.3.26)

˜ t (φ)| = 0 for all φ ∈ D0 , lim sup |Lεt n (φ) − L

n→∞ t≤n

and (recall Corollary III.1.7) Rδ ≡ cl(∪t≥δ S(Zt1 ) ∪ S(Zt2 )) is compact for all δ > 0.   2 : x ∈ R Let η > 0. The definition of Lε shows that Kδ = x1 +x is a compact i δ 2 ε ε ˜ · (1) is continuous and allows support for L∞ (X) − Lδ (X). Our choice of ω implies L us to choose δ > 0 so that Lεδn (1) < η for all n. Therefore (IV.3.27)

Lε∞n (X)(Kδc ) = Lεδn (Kδc ) < η

for all n.

Therefore {Lεt n (X) : n ∈ N, t ≥ 0} are tight and (IV.3.26) shows that for each t ≥ 0, all limit points of {Lεt n } in the weak topology on MF (Rd ) coincide. Therefore there is an MF (Rd )-valued process Lt (X) such that limn→∞ Lεt n (X) = Lt (X) for all t ≥ 0 a.s., Lt (X) is non-decreasing in t and satisfies (IV.3.28)

˜ t (φ) for all t ≥ 0 and φ ∈ D0 a.s. Lt (X)(φ) = L

In particular Lt (X)(φ) is continuous in t ≥ 0 for all φ ∈ D0 a.s. and hence Lt (X) bp

is a.s. continuous in t as well. If ψn →ψ, then using Dominated Convergence in (IV.3.24) one can easily show there is a subsequence such that ˜ t (ψn ) = L ˜ t (ψ) ∀t ≥ 0 a.s. lim L k

k→∞

by showing this is the case for each of the other terms in (IV.3.24). (A subsequence is needed as one initially obtains L2 convergence for the martingale terms.) It then follows from (IV.3.28) that (IV.3.25) holds and the proof is complete. Proof of Theorem IV.3.2. (a) As we may take X = Z in Theorem IV.3.8, it remains only to show that Lt (Z) is not identically 0. The L2 convergence in Theorem IV.3.8 and a simple second moment calculation show that 2t 1 ps (z1 − z2 )ds m1 (dz1 )m2 (dz2 ) = 0. E(Lt (Z)(1)) = lim E(Lεt (Z)(1)) = ε↓0 2 0 (b) We first give a careful argument if d > 6. Recall the definition of Gδ (X) given at the beginning of this Section and recall that h(u) = (u log+ (1/u))1/2 . If δi (3, ω) is as in Corollary III.1.5, then that result and the fixed time hitting estimate, Theorem III.5.11, show that for x ∈ Rd and t > 0, P(Zsi (B(x, ε)) > 0 for some s ∈ [t, t + ε2 (log+ (1/ε))−1 ], and δi (3, ω) > ε2 (log+ (1/ε))−1 ) ≤ P(Zti (B(x, ε + 3h(ε2 (log+ (1/ε))−1 ))) > 0)  d−2 ≤ Cd γi−1 t−d/2 mi (1) ε + 3h(ε2 (log+ (1/ε))−1 ) (IV.3.29)

≤ Cd γi−1 t−d/2 mi (1)εd−2 .

274

Superprocesses

Let Sn = {B(xni , 2−n ) : 1 ≤ i ≤ cd nd 2nd } be an open cover of [−n, n]d . If δ > 0, Rδ is as in (IV.3.27), and ηn = 2−2n (log 2n )−1 , then P(Gδ (Z 1 ) ∩ Gδ (Z 2 ) = ∅, Rδ ⊂ [−n, n]d , Zn1 = Zn2 = 0, δ1 (3) ∧ δ2 (3) > ηn , )   ≤ P(Zs1 (B(xni , 2−n ))Zs2 (B(xni , 2−n )) > 0 −1 1≤i≤c nd 2nd 0≤j≤nηn d

for some s ∈ [δ + jηn , δ + (j + 1)ηn ], δ1 (3) ∧ δ2 (3) > ηn ) ≤

(nηn−1

+ 1)cd n 2 (Cd )2 (γ1 γ2 )−1 δ −d m1 (1)m2 (1)2−n2(d−2) d nd

≤ c(d, δ)m1 (1)m2 (1)n

2+d −n(d−6)

2

by (IV.3.29)

→ 0 as n → ∞.

As n → ∞ the left-hand side of the above approaches P(Gδ (Z 1 ) ∩ Gδ (Z 2 ) = ∅) by Corollary III.1.5 and (IV.3.27), and so the result follows by letting δ ↓ 0. Finally we sketch the argument in the critical 6-dimensional case. First (IV.3.29) can be strengthened to (IV.3.30)

P



t+ε2

 Zsi (B(x, ε)) ds > 0 ≤ cd t−d/2 mi (1)εd−2

∀t ≥ 4ε2 , d ≥ 3.

t

This may shown using an appropriate nonlinear pde as in Section III.5. A short proof is given in Proposition 3.3 of Barlow, Evans and Perkins (1991). Now introduce a restricted Hausdorff measure q f (A) for A ⊂ R+ × Rd and f : [0, ε) → R+ a nondecreasing function for which f (0+) = 0. It is given by q f (A) = lim inf δ↓0

∞ 

2 f (ri ) : A ⊂ ∪∞ i=1 [ti , ti + ri ] ×

i=1

d ,

 [xji , xji + ri ], ri < δ .

j=1

If d > 4 and ψd (r) = r 4 log log(1/r) (as in Theorem III.3.9) then there are 0 < c1 (d) ≤ c2 (d) < ∞ so that ∞ 1A (s, x)Zsi (dx) ds ≤ c2 q ψd (A ∩ G(Z i )) c1 q ψd (A ∩ G(Z i )) ≤ 0

(IV.3.31)

∀A ∈ B(R+ × Rd ) a.s.

This is Theorem 3.1 of Barlow, Evans, and Perkins (1991) and may be shown using the ideas presented in Section III.3. (It should be possible to prove c1 = c2 here.) d If q d = q r , then a simple consequence of (IV.3.30) (cf. Corollary III.5.10) is (IV.3.32)

q d−2 (A) = 0 implies A ∩ G(Z 1 ) = ∅ a.s. for all A ⊂ R+ × Rd , d ≥ 3.

(IV.3.31) shows that q d−2 (G(Z 2 )) = 0 if d ≥ 6 and so (IV.3.32) with A = G(Z 2 ) implies that G(Z 1 ) ∩ G(Z 2 ) = ∅ a.s. Proof of Lemma IV.3.3. If d = 2 this is a simple consequence of Theorem III.3.4, so assume d > 2. We may assume that our space carries independent historical processes (H 1 , H 2 ) associated with (Z 1 , Z 2 ). Let h and δi (3, H i ) be as in

Superprocesses

275

¯ d and r0 (δ, H 1 ) be the Historical Modulus of Continuity (Theorem III.1.3) and let h as in Theorem III.3.4. Those results show that on {ω : δi (3, H i ) > 2−n , i = 1, 2, and r0 (δ, H 1 ) > h(2−n )}, we have

sup

t≥δ,t∈[j2−n ,(j+1)2−n ]

≤



sup

t≥δ,t∈[j2−n ,(j+1)2−n ]

≤

1(|z1 − z2 | ≤ h(2−n ))Zt1 (dz1 )Zt2 (dz2 )

sup

t∈[j2−n ,(j+1)2−n ]

1(|y1 (j2−n ) − y2 (j2−n | ≤ 7h(2−n ),

|y1 (t) − y2 (t)| ≤ h(2−n ))Ht1 (dy1 )Ht2 (dy2 ) ¯ d (h(2−n )) γ1 c(d)h   1 7h(2−n ) Ht2 (dy2 ). × 1 y2 (j2−n ) ∈ S(Zj2 −n )

(IV.3.33)

A weak form of Lemma III.1.6 (with s fixed) has also been used in the last line. If H 1,∗ = supt Ht1 (1), then (III.3.1) and the ensuing calculation show that for n ≥ N (H 1 ), 1 7h(2 S(Zj2 −n )

−n

)

(IV.3.34) ⊂ a union of γ1−1 (H 1,∗ + 1)2n+2 balls of radius 10h(2−n ) ∀j ∈ N. 2 1 7h(2 Let Wn (j) = Zj2 −n (S(Zj2−n ) 1 n ≥ N (H ). Then (IV.3.34) implies

1 7h(2 P x (Bs ∈ S(Zj2 −n )

(IV.3.35) Therefore

−n

0

∞

Condition on H 1 and assume that

).

−n

) ≤ cd γ1−1 (H 1,∗ + 1)2n h(2−n )d s−d/2 .

−n

1 7h(2 sup P x (Bs ∈ S(Zj2 −n ) x ≤ (cd γ1−1 + 1)(H 1,∗ + 1)

≤ c(d, γ1 )(H 1,∗ + 1)2 ≡

)

∞

) ds min(2n h(2−n )d s−d/2 , 1) ds

0 −n(1−2/d)

log 2n

γ2−1 λ−1 n . −n

1 7h(2 ) ) and G(fn , t) is as in Lemma III.3.6, then If fn (x) = λn 1(x ∈ S(Zj2 −n ) −n γ2 G(fn , j2 ) ≤ 1 and so Lemma III.3.6 implies that on {n ≥ N (H 1 )} and for j2−n ≥ 1/n, 1 −17n E(eλn Wn (j) |H 1 ) P(Wn (j) ≥ 17nλ−1 n |H ) ≤ e   1 7h(2−n ) ≤ e−17n exp m2 (1)2λn sup P x (Bj2−n ∈ S(Zj2 ) −n ) x

(IV.3.36)

−17n

≤e



exp(m2 (1)c (δ, γ1 , γ2 ))

(by (IV.3.35)).

276

Superprocesses

The Markov property and (III.1.3) show that conditional on 2 −n 1 7h(2−n ) σ(H 1 ) ∨ σ(Hs2 , s ≤ j2−n ), t → Hj2 ) ∈ S(Zj2 }) is equal −n +t ({y : y(j2 −n ) 2 −1 in law to PWn (j)δ0 (Z· (1) ∈ ·). Therefore if ηn > 0 and Kn = 17nλn , then   1 7h(2−n ) Ht2 ({y : y(j2−n ) ∈ S(Zj2 }) > ηn , Wn (j) < Kn |H 1 P sup −n ) t∈[j2−n ,(j+1)2−n ]

    ≤ E PWn (j)δ0 sup exp(2n γ2−1 Zt2 (1)) > exp(2n ηn γ2−1 ) 1(Wn (j) < Kn )|H 1 t≤2−n

≤ exp(−2

n

  EKn δ0 (exp(2n γ2−1 Z22−n (1)))|H 1 (weak L1 inequality)

ηn γ2−1 )E

≤ exp(−2n ηn γ2−1 )E(exp(Kn 2n+1 /γ2 )|H 1 )

(Lemma III.3.6).

Set ηn = 35n/λn = c (d, γ1 , γ2 )(H 1,∗ + 1)2−n(1−2/d) n2 and use (IV.3.36) in the above to conclude that on {n ≥ N (H 1 )} and for j2−n ≥ 1/n,   1 7h(2−n ) Ht2 ({y : y(j2−n ) ∈ S(Zj2 }) > ηn |H 1 P sup −n ) t∈[j2−n ,(j+1)2−n ]

 2n η (34)2n n  n ≤ e−17n exp(m2 (1)c (δ, γ1 , γ2 )) + exp − + γ2 γ2 λn  2n n  ≤ e−17n exp(m2 (1)c (δ, γ1 , γ2 )) + exp − γ2 λn ≤ e−17n exp(m2 (1)c (δ, γ1 , γ2 )) + exp(−c(d, γ1 )22n/d (log 2)n2 ). A conditional application of Borel-Cantelli now shows there is an N (H) < ∞ a.s. so that for n ≥ N (H), sup

j2−n ≥1/n

sup

t∈[j2−n ,(j+1)2−n ]

1 7h(2 Ht2 ({y : y(j2−n ) ∈ S(Zj2 −n )

−n

)

})

≤ c (δ, γ1 , γ2 )(H 1,∗ + 1)2−n(1−2/d) n2 . Use this in (IV.3.33) to see that for a.a. ω if n is sufficiently large, then 1(|z1 − z2 | ≤ h(2−n ))Zt1 (dz1 )Zt2 (dz2 ) sup t≥δ

¯ d (h(2−n ))2−n(1−2/d) n2 . ≤ γ1 c(d)c (d, γ1 , γ2 )(H 1,∗ + 1)h An elementary calculation now completes the proof. 4. A Singular Competing Species Model–Higher Dimensions. In this Section we describe how to use collision local time to formulate and solve the competing species model introduced in Section IV.1 in higher dimensions. The actual proof of the main results (due to Evans and Perkins (1994,1998) and Mytnik (1999)) are too long to reproduce here and so this Section will be a survey of known results together with some intuitive explanations. We use the notation of Section IV.1 with Ei = Rd and Ai = ∆/2. In particular, 2 Ω = C(R+ , MF (Rd ))2 with its Borel σ-field F 2 and canonical right-continuous

Superprocesses

277

filtration Ft2 . In view of Remark IV.3.1(b), here is the natural extension of (CS)λm to higher dimensions. Definition. Let λ = (λ1 , λ2 ) ∈ R2+ and m = (m1 , m2 ) ∈ MF (Rd )2 . A probability P on (Ω2 , F 2 ) satifies (GCS)λm iff t  ∆φi  i Xsi Xt (φi ) = mi (φi ) + ds + Mti (φi ) − λi Lt (X)(φi ), 2 0 (GCS)λm

where Mti (φi ) is a continuous Ft2 − martingale under P such that t Xsi (φ2i )ds M0i (φi ) = 0 andM i (φi ), M j (φj )t = δij 0

for φi ∈ D(∆/2) and i = 1, 2. The existence of Lt (X) is implicit in (GCS)λm . We will say X = (X1 , X2 ) satisfies (GCS)λm if X is a process whose law satisfies (GCS)λm . Let (GCS)ε,λ m denote the corresponding martingale problem in which Lt (X)(φi ) is replaced by Lε,i t (X)(φi ) for i = 1, 2 (recall Remark IV.3.1(c)). Note first that the Domination Principle (Proposition IV.1.4) shows that if X satisfies (CS)λm , we may assume there are a pair of independent super-Brownian motions (Z 1 , Z 2 ) such that X i ≤ Z i a.s. If d ≥ 6, then Theorem IV.3.2(b) implies G(X 1 ) ∩ G(X 2 ) ⊂ G(Z 1 ) ∩ G(Z 2 ) = ∅ and so L(X) must be 0 a.s. Corollary IV.1.3 (with gi0 = 0) now shows that (X 1 , X 2 ) is a pair of independent superBrownian motions. Conversely, using Theorem IV.3.2(b) it is easy to see that a pair
of independent super-Brownian motions does satisfy (GCS)λm with L(X) = 0 if gd−2 (|z1 − z2 |)m1 (dz1 )m2 (dz2 ) < ∞. (The latter condition ensures that supε>0 E(Lεδ (X)(1)) approaches 0 as δ ↓ 0, and Theorem IV.3.2(b) shows that a.s. Lεt (X)(1) − Lεδ (X)(1) → 0 as ε ↓ 0 for any δ > 0.) Therefore we only consider the above martingale problem for d ≤ 5 when non-trivial solutions may exist. Next we show that if d = 1, then (GCS)λm may be viewed as a generalization of (CS)λm . 0 of (CS)λm Proposition IV.4.1. Assume d = 1 and m ∈ F . The unique solution Pm λ also satisfies (GCS)m . Proof. We need only show that  t  u1 (s, x)u2 (s, x)ds dx P0m a.s. (IV.4.1) Lt (X)(dx) = 0

Let φ ∈ Cb (R). Theorem IV.2.1 shows that Xsi (dx) = ui (s, x)dx for all s > 0 P0m -a.s. and Proposition IV.2.3 shows that t → ui (t, ·) is a continuous map from (0, ∞) to CK (R) P0m -a.s. It is now easy to see that P0m -a.s. for all 0 < δ ≤ t, limLεt (X)(φ) − Lεδ (X)(φ) ε↓0 t x1 + x 2 )pε (x1 − x2 )u1 (s, x1 )u2 (s, x2 )dx1 dx2 ds (IV.4.2) = lim φ( ε↓0 δ 2 t φ(x)u1 (s, x)u2 (s, x)dxds. = δ

278

Superprocesses

Note also by the Domination Principle, δ   0 ε Pm Lδ (X)(1) + u1 (s, x)u2 (s, x)dxds (IV.4.3)

0 δ

p2s+ε (x1 − x2 ) + p2s (x1 − x2 )dsdm1 (x1 )dm2 (x2 ) √ 0 ≤ c δm1 (1)m2 (1) → 0 as δ ↓ 0. ≤

(IV.4.1) now follows from (IV.4.2) and (IV.4.3). ε,λ Recall that Pεm is the unique solution of (CS)ε,λ m which is equivalent to (GCS)m . ε In view of Remark IV.3.1(c) we may expect Pm to converge to a solution of (GCS)λm as ε ↓ 0.
1 Notation. MF S (Rd ) = {m ∈ MF (Rd ) : 0 r 1−d supx m(B(x, r))dr < ∞}.

If m1 , m2 ∈ MF S , then an integration by parts shows that sup gd−2 (|z1 − z2 |)dm1 (z1 ) < ∞ z2

and so (m1 , m2 ) satisfies the hypothesis (IC) of Theorem IV.3.8. Theorem IV.4.2. (a) Assume 1 ≤ d ≤ 3 and m ∈ (MF S )2 . w (i) Then Pεm ⇒Pm on Ω2 , where ((Xt )t≥0 , (Pν )ν∈MF S ) is an (MF S )2 -valued Borel (Ft2 )-strong Markov process and Pm satisfies (GCS)λm . (ii) If, in addition, λ1 = λ2 . then Pm is the unique solution of (GCS)λm . (b) If d = 4 or 5, m ∈ (MF (Rd ) − {0})2 satisfies the hypothesis (IC) of Theorem IV.3.2, and λ = (0, 0), then there is no solution to (GCS)λm . Discussion.(b) Theorem IV.3.8 shows that the existence of a collision local time for for any potential solutions of (GCS)λm is to be expected if d ≤ 5 and Theorem IV.3.2 suggests it will be nontrivial for d ≤ 5. These results may lead one to believe that nontrivial solutions exist for d ≤ 5. It turns out, however, that it is not the existence of collisions between a pair of independent super-Brownian motion that is germane to the existence of the solutions to (GCS). Rather it is the existence of collisions between a single Brownian path, B, and an independent super-Brownian motion, Z. If G(B) = {(t, Bt ) : t ≥ 0}, then (IV.4.4)

P(G(B) ∩ G(Z) = ∅) > 0 iff d < 4.

To see this for d ≥ 4, recall from (IV.3.31) that q ψd (G(Z)) < ∞ a.s. We had d > 4 there but the proof in Theorem 3.1 of Barlow, Evans and Perkins (1991) also goes through if d = 4. This shows that q d (G(Z)) = 0 if d ≥ 4 and so (IV.4.4) is true by Theorem 1 of Taylor and Watson (1985) (i.e., the analogue of (IV.3.32) for G(B)). For d ≤ 3 one approach is to use a Tanaka formula to construct a nontrivial inhomogeneous additive functional of B which only increases on the set of times when B(t) ∈ S(Zt ) (see Theorem 2.6 of Evans-Perkins (1998)). The construction requires a mild energy condition on the initial distributions of B and Z but the required result then holds for general initial conditions by Theorem III.2.2. Alternatively, a

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short direct proof using Theorem III.3.4 is given in Proposition 1.3 of Barlow and Perkins (1994). To understand the relevance of (IV.4.4), we demonstrate its use in a heuristic proof of (b). Assume X satisfies (GCS)λm for d = 4 or 5. Let Z i ≥ X i be a pair of dominating independent super-Brownian motions (from Proposition IV.1.4) and let H i be the historical process associated with Z i . The particle approximations in Example IV.1.1 suggest that X 1 is obtained from Z 1 by killing off some of the particles which collide with the X 2 population, and similarly for X 2 . Use the notation of the Historical Cluster Representation (Theorem III.1.1) and let {y1 , . . . , yM } be the finite support of rt−ε (Ht1 ) for fixed 0 < ε < t. These are the ancestors at time t − ε of the entire Z 1 population at time t. Which of these ancestors are still alive in i (·))/m1 (1) the X 1 population at time t − ε? By Theorem III.3.1, yi has law E(Ht−ε and so is a Brownian path stopped at time t − ε and is independent of Z 2 . (IV.4.4) shows that G(yi ) ∩ G(Z 2 ) = ∅ a.s. Therefore each yi will not have encountered the smaller X 2 population up to time t − ε and so must still be alive in the X 1 population. Let ε ↓ 0 to see that the entire family tree of the population of Z 1 at time t never encounters Z 2 and hence X 2 . This means that no particles have been killed off and so Zt1 = Xt1 a.s., and by symmetry, Zt2 = Xt2 a.s. These identities hold uniformly in t a.s. by continuity. The fact that P(L(Z) = 0) > 0 (Theorem IV.3.2) shows that Z does not satisfy (GCS)λm and so no solution can exist. In short, for d = 4 or 5, the only collisions contributing to Lt (Z) are between particles whose family trees die out immediately and so killing off these particles has no impact on the proposed competing species model. The above proof is not hard to make rigorous if there is a historical process associated with X i so that we can rigorously interpret the particle heuristics. To avoid this assumption, the proof given in Section 5 of Evans and Perkins (1994) instead uses the ideas underlying the Tanaka formula in the previous Section. The proof outlined above would also appear to apply more generally to any killing operation based on collisions of the two populations. In (GCS) we would replace λi Lt (X)(φi ) with Ait (φi ), where Ai· is an increasing continuous MF -valued process such that S(Ai (dt, dx)) ⊂ G(X 1 ) ∩ G(X 2 ) a.s. The non-existence of solutions for d = 4 or 5 in this more general setting is true (unpublished notes of Barlow, Evans and Perkins) but the 4-dimensional case is rather delicate. (a) Tightness of {Pεm } is a simple exercise using the Domination Principle and Theorem IV.3.2. To show each limit point satisfies (GCS)λm , a refinement of Theorem IV.3.8 is needed for d ≤ 3 (see Theorem 5.10 of Barlow, Evans and Perkins (1991)). This refinement states that in (IV.3.10) the rate of convergence to 0 in probability is uniform in X satisfying (H2 ). In the proof of (IV.3.10), the only step for which this additional uniformity requires d ≤ 3 (and which requires some serious effort) is (IV.3.14). To handle this term we use Theorem III.3.4 to first bound the integrals w with respect to Xsi (dxi ) at least if d ≤ 3. If Pεmn ⇒, use Skorohod’s theorem to obtain εn εn ,λ solutions X of (CS)m which converge a.s. to X, say, as n → ∞. We now may let n → ∞ in (CS)εmn ,λ to derive (GCS)λm for X–the above uniformity and a simple comparison of Lεn (X εn ) with Lεn ,i (X εn ) (see Lemma 3.4 of Evans and Perkins (1994)) show that Lεn ,i (X εn ) → L(X) in probability as n → ∞ and the other terms are easy to handle.

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To proceed further seems to require considerable additional effort. The full ε } to a nice strong Markov process is provided in Evans and convergence of the {Pm Perkins (1998) (Theorems 1.6 and 8.2). Here we showed that each limit point has an associated pair of historical processes which satisfy a strong equation driven by a pair of independent historical Brownian motions whose supports carry a Poisson field of marks indicating potential killing locations. This strong equation has a solution which is unique, both pathwise and in law. (This general approach of using strong equations driven by historical processes will be used in another setting with greater attention to detail in the next Section.) This approach does show that the natural historical martingale problem associated with (GCS)λm is wellposed (Theorem 1.4 of Evans and Perkins (1998)). The uniqueness of solutions to (GCS)λm itself remains open in general as we do not know that any solution comes equipped with an associated historical process (from which we would be able to show it is the solution of the aforementioned strong equation). If λ1 = λ2 , uniqueness of solutions to (GCS)λm was proved by Mytnik (1999) by a duality argument. Mytnik built a dual family of one-dimensional distributions (as opposed to a dual process) by means of an intricate and original Trotter product construction. One phase of the Trotter product requires solutions to a non-linear evolution equation with irregular initial data. As is often the case with duality arguments, it is non-robust and does not appear to handle the case where λ1 = λ2 . It is somewhat disconcerting that after all of this effort the general question of uniqueness to our competing species model remains unresolved in general. I suspect the correct approach to these questions remains yet to be discovered and so was not tempted to provide a detailed description of the proofs here.

V. Spatial Interactions 1. A Strong Equation We continue our study of measure-valued processes which behave locally like (A, γ, g)-DW superprocesses, i.e., where A, γ, and g may depend on the current state, Xt , of the process. In this Chapter we allow the generator governing the spatial motion, A, to depend on Xt . These results are taken from Perkins (1992), (1995). To simplify the exposition we set γ = 1 and g = 0, although as discussed below (in Section V.5) this restriction may be relaxed. Our approach may be used for a variety of dependencies of AXt on Xt but we focus on the case of state dependent diffusion processes. Let σ : MF (Rd ) × Rd → Rd×d , and set Aµ φ(x) =

 i

b : MF (Rd ) × Rd → Rd ,

aij (µ, x)φij (x) +

j



bi (µ, x)φi (x),

a = σσ ∗ ,

for φ ∈ Cb2 (Rd ).

i

Here a(µ, x) and b(µ, x) are the diffusion matrix and drift of a particle at x in a population µ. If Lip1 = {φ : Rd → R : φ∞ ≤ 1, |φ(x) − φ(y)| ≤ x − y ∀x, y ∈ Rd } and µ, ν ∈ MF (Rd ), the Vasershtein metric on MF (Rd ), introduced in Section II.7, is d(µ, ν) = sup{|µ(φ) − ν(φ)| : φ ∈ Lip1 }. Recall that d is a complete metric on MF (Rd ) inducing the topology of weak convergence. Our approach will be based on a fixed point argument and so we will need the following Lipschitz condition on b and σ: Assume there is a non-decreasing function L : R+ → R+ such that (a) σ(µ, x) − σ(µ , x ) + b(µ, x) − b(µ , x ) (Lip) ≤ L(µ(1) ∨ µ (1))[d(µ, µ) + x − x ] ∀µ, µ ∈ MF (Rd ), x, x ∈ Rd . (b) sup σ(0, x) + b(0, x) < ∞. x

Remark V.1.1. (a) (Lip) easily implies that for some non-decreasing C : R+ → R+ , (B)

σ(µ, x) + b(µ, x) ≤ C(µ(1)) ∀µ ∈ MF (Rd ), x ∈ Rd .

(b) The results of Sections V.1-V.4 remain valid without (Lip)(b) (see Section 5 of Perkins (1992)).

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Exercise V.1.1. Prove that (Lip) holds in the following cases. (a) σ(µ, x) = f (µ(φ1 ), . . . , µ(φn ), x), where φi are bounded Lipschitz functions on d×d is Lipschitz continuous so that supx f (0, x) < ∞. Rd and f : Rn+d n →
R (b) b(µ, x) = k=1
bk (x, x1 , . . . , xk )dµ(x1 ) . . . dµ(xk ) n σ(µ, x) = k=1 σk (x, x1 , . . . , xk )dµ(x1 ) . . . dµ(xk ), where bk and σk are bounded Lipschitz continuous functions taking values in Rd and Rd×d , respectively.
A special case of (b) would be b(µ, x) = b1 (x, x1 )dµ(x1 ) and
σ(µ, x) = pε (x − x1 )dµ(x1 ). Here b1 (x, x1 ) ∈ Rd models an attraction or repulsion between individuals at x and x1 , and particles diffuse at a greater rate if there are a number of other particles nearby. To motivate our stochastic equation, consider the branching particle system in d Section II.3 where Y α ≡ B α are Brownian motions in Rd , X0 = m ∈ MF (R ), 1 1 1 n N α and ν = ν = 2 δ0 + 2 δ2 . Recall from (II.8.3) that if Ht = N α∼t δB·∧t , then H N converges weakly to a historical Brownian motion, H, with law Q0,m . Let Z0 : Rd → Rd be Borel. Now solve t t |α| + 1 (SE)N (a) Ztα = Z0 (B0α ) + σ(XsN , Zsα )dBsα + b(XsN , Zsα )ds, t < N 0 0 1  N (b) Xs = δZsβ . N β∼s

Such solutions are easy to construct in a pathwise unique manner on [i/N, (i+1)/N ) by induction on i. On [i/N, (i + 1)/N ), we are solving a finite system of stochastic differential equations driven by {Bsα : α ∼ i/N, s ∈ [i/N, (i + 1)/N )} and with Lipschitz continuous coefficients. The latter uses  1   1  d δxα , δxˆα ≤ N −1 xα − x ˆα  N N α∼i/N

α∼i/N

α∼i/N

≤ where x− x ˆ 2 =



α ˆ α 2 α∼i/N x − x

N (1))−1/2 N −1/2 x (Hi/N

−x ˆ 2 ,

1/2 and we have used Cauchy-Schwarz in the

last inequality. This shows there is a pathwise solution to (SE)N on [i/N, (i+1)/N ). Now let the B α ’s branch at t = (i+1)/N and continue on [(i+1)/N, (i+2)/N ) with the new set of Brownian motions {Bsα : α ∼ (i + 1)/N, s ∈ [(i + 1)/N, (i + 2)/N )}. These solutions are then pieced together to construct the {Ztα : t < (|α| + 1)/N, α} w in (SE)N . If N → ∞, we may expect X N ⇒X, where t t (SE) σ(Xs , Zs )dy(s) + b(Xs , Zs )ds (a) Zt (ω, y) = Z0 (y0 ) + 0 0 (b) Xt (ω)(A) = 1(Zt (ω, y) ∈ A)Ht (ω)(dy) ∀A ∈ B(Rd ) The intuition here is that ω labels a tree of branching Brownian motions and y labels a branch on the tree. Then Zt (ω, y) solves the sde along the branch y in the tree ω and Xt (ω) is the empirical distribution of these solutions. Our objective in this

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Chapter is to give a careful interpretation of the stochastic integral in (SE)(a), prove w that (SE) has a pathwise unique strong Markov solution and show that X N ⇒X. 2. Historical Brownian Motion Throughout this Section we work in the setting of the historical process of Section II.8 where (Y, P x) ≡ (B, P x) is d-dimensional Brownian motion. We adopt the notation given there with E = Rd , but as B has continuous paths we replace (D(E), D) with (C, C), where C = C(R+ , Rd ) and C is its Borel σ-field. Let Ct = σ(ys , s ≤ t) be its canonical filtration. If Z : R+ × C → R, then Z is (Ct )-predictable ⇐⇒ Z is (Ct )-optional ⇐⇒ Z is Borel measurable and Z(t, y) = Z(t, y t ) ∀t ≥ 0.

(V.2.1)

This follows from Theorem IV.97 in Dellacherie and Meyer (1978) and the fact that the proofs given there remain valid if D is replaced by C. We will therefore identify ˆ d = {(t, y) ∈ R+ × C : y = y t } with (Ct )-predictable functions Borel functions on R s on R+ × C. If C = {y ∈ C : y = y s } then this identification allows us to write the domain of the weak generator for the path-valued process W in (II.8.1) as ˆ = {φ : R+ × C → R : φ is bounded, continuous, and (Ct )-predictable, D(A) and for some Aˆs φ(y) with the same properties, φ(t, B) − φ(s, B) t Aˆr φ(B)dr, t ≥ s is a (Ct )-martingale under Ps,y ∀s ≥ 0, y ∈ C s }. − s

y ∈ C s, Recall here that Ps,y is Wiener measure starting at time s with past history
s s and for m ∈ MF (C) (recall this means y = y m− a.s.) define Ps,m = Ps,y m(dy). For the rest of this Section assume τ ≥ 0 and (Kt )t≥τ satisfies (HM P )τ,Kτ (from Section II.8) on (Ω, F , (Ft)t≥τ , P) with Kτ now possibly random with law ν, γˆ ≡ 1, gˆ ≡ 0, and Aˆ equal to the generator of the path-valued Brownian motion described above. Assume that this probability space is complete, the filtration is right-continuous and Fτ contains all the null sets. We also assume E(Kτ (1)) < ∞ so that m(·) = E(Kτ (·)) ∈ MFτ (C) and we can still work with a martingale problem as opposed to a local martingale problem. Call such a process, K, an (Ft )-historical Brownian motion starting at (τ, ν) (or (τ, m) if ν = δ
m ). As in Theorem II.8.3, K is an (Ft )-strong Markov process and has law Qτ,ν = Qτ,K0 dν(K0 ). In this setting the superprocess property (II.8.5) becomes (V.2.2)

P(Kt (ψ)) = Pτ,m (ψ(B t)) for t ≥ τ, ψ ∈ bC.

Note also that if S ≥ τ is a finite valued (Ct )-stopping time, then (V.2.3)

Pτ,m (g|CS )(y) = PS,yS (g) Pτ,m − a.a. y ∀g ∈ bC.

To see this write g(y) = g˜(y S , y(S + ·)) and use the strong Markov property at time S. Our main objective in this Section is the seemingly minor extension of (HM P )τ,Kτ presented in Proposition V.2.6 below, and the reader may want to skip ahead to this result and its Corollary V.2.7. The latter plays a key role in what

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follows. Note, however, that Proposition V.2.4 will also be used in our stochastic calculus on Brownian trees and the proof of Lemma V.2.2 illustrates a neat idea of Pat Fitzsimmons. ˆ to zero starting times. We first reduce the definition of D(A) Notation. bP is the space of bounded Ct -predictable processes on R+ × C. ˆ iff φ ∈ bP is continuous and for some continuous Lemma V.2.1. φ ∈ D(A) ˆ ∈ bP, Aφ t Aˆr φ(y)dr is a (Ct )-martingale under P x ∀x ∈ Rd . n(t, y) = φ(t, y) − φ(0, y) − 0

Proof. We need only show the above condition is sufficient for membership in ˆ Assume φ is as above and let s > 0. It suffices to show n(t) − n(s), t ≥ s is a D(A). (Ct )t≥s -martingale under Ps,y for every y ∈ C s . Let t ≥ r ≥ s and ψ be a bounded continuous Cr -measurable mapping on C. We must show that Ps,y (n(t)ψ) = Ps,y (n(r)ψ) ∀y ∈ C s .

(V.2.4) The left-hand side is (V.2.5)

P 0 (n(t, y/s/(y(s) + B))ψ(y/s/(y(s) + B)))

and so is continuous in y by Dominated Convergence. The same is true of the righthand side. It therefore suffices to establish (V.2.4) on a dense set of y in C s . Next we claim that (V.2.6)

the closed support of P y0 (B s ∈ ·) is {y ∈ C s : y(0) = y0 }.

To see this first note that for every ε, T > 0, P y0 (sups≤T |Bs − y0 | < ε) > 0 (e.g. by the explicit formula for the two-sided hitting time in Theorem 4.1.1 of Knight (1981)). Now use the classical Cameron-Martin-Girsanov formula to conclude that for any ψ ∈ C(R+ , Rd ), and ε, T > 0, s ψ(u)du − y0 | < ε) > 0. P y0 (sup |Bs − s≤t

0

The claim follows easily. It implies that (V.2.4) would follow from Ps,ys (n(t)ψ) = Ps,ys (n(r)ψ) P y0 − a.a. y for all y0 ∈ Rd . By (V.2.3) this is equivalent to P y0 (n(t)ψ|Cs )(y) = P y0 (n(r)ψ|Cs )(y) P y0 − a.a. y for all y0 ∈ Rd . This is immediate by first conditioning n(t)ψ with respect to Cr . Fitzsimmons (1988) showed how one can use Rost’s theorem on balayage to establish sample path regularity of a general class of superprocesses. Although we have not needed this beautiful idea for our more restrictive setting, the next result illustrates its effectiveness.

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Lemma V.2.2. Let φ, ψ : R+ × C → R be (Ct )-predictable maps such that for some fixed T ≥ τ , φ(t, y) = ψ(t, y) ∀τ ≤ t ≤ T Pτ,m − a.s. Then φ(t, y) = ψ(t, y) Kt − a.a. y

∀τ ≤ t ≤ T

P − a.s.

Proof. Return to the canonical setting of historical paths, (Ω, FH , F H [τ, t+], Qτ,ν ) ˆ of Section II.8, with gˆ = 0, γˆ = 1, and P x =Wiener measure. Recall the E-valued τ +t ˆ ) with laws Pτ,y and the W -superprocess diffusion Wt = (τ + t, Y (V.2.7)

ˆ τ,m . ˆ t = δτ +t × Hτ +t with laws P X

Note first that (V.2.8) if g : [τ, ∞)×C → R+ is (Ct )t≥τ −predictable then Ht (gt ) is F H [τ, t+]−predictable. To see this start with g(t, y) = g1 (t)g2 (y t ), where g1 , g2 are non-negative bounded continuous functions on R+ and C, respectively. Then (V.2.8) holds because Kt (gt ) is a.s. continuous. A monotone class argument now proves (V.2.8) (recall (V.2.1)). Let S be an F H [τ, t+])t≥τ -stopping time such that τ ≤ S ≤ T and let λ > 0. ˆ ˆ by Then Sˆ = S − τ is an (FtX )t≥0 -stopping time. Define a finite measure µ on E ˆ ˆ (g)), ˆ τ,ν (e−λSˆ X µ(g) = Qτ,ν (e−λ(S−τ ) HS (gS )) = P S where the second equality holds by Lemma II.8.1. Let Uλ f be the λ-resolvent of W . ˆ then the superprocess property ((II.8.5) and If f is a non-negative function on E, the display just before it) shows that   ∞ ˆ τ,ν ˆ t (f ) dt e−λt X δτ × m, Uλ f  = P 0 ∞   −λ ˆ τ,ν e Sˆ ˆ ˆ (f )dt ≥P e−λt X t+S 0   ∞  ˆˆ −λS ˆ ˆ t (f )dt e−λt X PXˆ ˆ = Pτ,ν e S

ˆ −λS

ˆ τ,ν (e =P

0

ˆ ˆ (Uλ f )) = µ, Uλ f . X S

A theorem of Rost (1971) shows there is a randomized stopping time, V , on ˆ × [0, 1] (i.e., V is jointly measurable and {y : V (y, u) ≤ t} ∈ Ct+ for C(R+ , E) ˆ all u ∈ [0, 1]) such that for every non-negative Borel function g on E, 1 Pˆτ,m (e−λV (u) g(WV (u) )) du (V.2.9) µ(g) = 0

≤

1

Pτ,m (g(τ + V (u), Y τ +V (u) )) du. 0

If g(t, y) = |φ(t, y)−ψ(t, y)|, then the right-hand side of (V.2.9) is zero by hypothesis and so HS (|φ(S) − ψ(S)|) = 0 a.s.

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The Section Theorem (Theorem IV.84 of Dellacherie and Meyer (1978)) and (V.2.8) then show that Ht (|φ(t) − ψ(t)|) = 0 ∀t ∈ [τ, T ] Qτ,ν − a.s. As K has law Qτ,ν , the result follows. Lemma V.2.3. Let n : [τ, ∞) × C → R be a (Ct )t≥τ -predictable L2 -martingale under Pτ,m . Then t Kt (nt ) = Kτ (nτ ) + n(s, y)dM (s, y) ∀t ≥ τ P − a.s. τ

and is a continuous square integrable (Ft )-martingale. Proof. Let N > τ . Then (V.2.3) and the Section Theorem imply that (V.2.10)

n(t, y) = Pt,yt (n(N )) ∀τ ≤ t ≤ N

Pτ,m − a.s.

Now let S = {X : C → R : X ∈ L2 (Pτ,m ), nX (t, y) ≡ Pt,yt (X) satisfies t X Kt (nX ) = K (n ) + nX (s, y)dM (s, y) ∀t ≥ τ P − a.s.}. τ t τ τ

Implicit in the above condition is that both sides are well-defined and finite. If X ∈ Cb (C), then nX is bounded and continuous on R+ × C (recall (V.2.5)) and nX is a continuous (Ct )-martingale under P x for all x ∈ Rd by (V.2.3). Lemma V.2.1 ˆ and An ˆ X = 0. (HM P )τ,K therefore shows that X ∈ S. shows that nX ∈ D(A) τ bp

Let {Xn } ⊂ S and assume Xn →X. Then Dominated Convergence shows that bp

X n nXn →nX , Kt (nX t ) → Kt (nt ) ∀t ≥ τ , and (use (V.2.2))   t (nXn (s, y) − nX (s, y))2Ks (dy)ds P τ t Pτ,m ((nXn (s, B s ) − nX (s, B s))2 )ds → 0 as n → ∞. = τ

Therefore we may let n → ∞ in the equation showing X n ∈ S to conclude that X ∈ S. This and Cb (C) ⊂ S show that bC ⊂ S. Let X be a non-negative function in L2 (Pτ,m ) and set Xn = X ∧ n ∈ S. X n Monotone Convergence shows that nXn ↑ nX ≤ ∞ pointwise and Kt (nX t ) ↑ Kt (nt ) for all t ≥ τ . (V.2.2) shows that    t  t P Pτ,m (nXn (s, B s ) − nX (s, B s ))2 ds (nXn (s, y) − nX (s, y))2Ks (dy)ds = τ τ t Pτ,m ((Xn − X)2 )ds (by (V.2.3)) ≤ τ

→ 0 as n → ∞.

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allows us to conclude that X ∈ S, as above. It also shows that
This t
X 2 n (s, y)dM (s, y) is a continuous L martingale. In addition we have τ X P(Kτ (nX τ )) = m(nτ ) = Pτ,m (X) < ∞. All these results now extend to any X in L2 (Pτ,m ) by considering the positive and negative parts of X. Taking X = n(N ) we obtain the required result for t ≤ N , but with n ˜ (t, y) = Pt,yt (n(N )) in place of n(t, y). Now use (V.2.10) and Lemma V.2.2 to obtain the required result for t ≤ N as none of the relevant quantities are changed for t ≤ N off a P-null set if we replace n ˜ with n. Finally let N → ∞. ˆ F, ˆ Fˆt ) = (Ω×C, F ×C, Ft ×Ct ) and let Fˆt∗ denote the universal Definition. Let (Ω, completion of Fˆt . If T is a bounded (Ft )t≥τ -stopping time (write T ∈ Tb and note this means that T ≥ τ ), the normalized Campbell measure associated with KT is ˆ T on (Ω, ˆ F) ˆ given by the probability P ˆ T (A × B) = P(1A KT (B))/m(1). P ˆ T , ω has law ˆ by (ω, y). Therefore under P We denote sample points in Ω −1 KT (1)m(1) dP and given ω, y is then chosen according to KT (·)/KT (1). We will also consider T ∈ Tb as an (Fˆt )-stopping time and define FˆT accordingly. Proposition V.2.4. (a) Assume T ∈ Tb and ψ ∈ bFˆT , then t ψ(y)dM (s, y) ∀t ≥ T

Kt (ψ) = KT (ψ) +

(V.2.11)

P-a.s.

T

ˆ be (Fˆt )-predictable and bounded on [τ, N ] × Ω ˆ for all N > τ . (b) Let g : [τ, ∞) × Ω Then   t s t t   gs (ω, y)dsKt(dy) = gr (ω, y)dr dM (s, y) + Ks (gs )ds (V.2.12) τ

τ

τ

τ

∀t ≥ τ

a.s.

Proof (a) Assume first T is constant and ψ(ω, y) = ψ1 (ω)ψ2 (y T ) for ψ1 ∈ bFT and ψ2 : C → R bounded and continuous. Then φ(s, y) = Ps,ys (ψ2 (B T )) is a bounded predictable (Ct )-martingale under P y0 for each y0 ∈ Rd (use (V.2.2)) ˆ and is continuous on R+ × C (as in (V.2.5)). Lemma V.2.1 shows that φ ∈ D(A) ˆ = 0. Therefore (HM P )τ,K implies that for t ≥ T , and Aφ τ t Kt (ψ) = ψ1 Kt (φt ) = ψ1 KT (ψ2 ) + ψ1

φ(s, y)dM (s, y) T

t = KT (ψ) +

ψ(y)dM (s, y), T

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because ψ1 φ(s) = ψ for s ≥ T . The proof now proceeds by a standard bootstrapping. The result clearly holds for ψ as above and T finite-valued and then for general T by the usual approximation of T by a decreasing sequence of finite-valued stopping times (the continuity of ψ2 helps here). A monotone class argument now gives the result for any ˜ y T ), where ψ˜ ∈ b(FT × C). We claim that any ψ ∈ bFˆT is ψ(ω, y) = ψ(ω, of this form. For any ψ ∈ bFˆT there is an (Fˆt )-predictable process X so that ψ = X(T ) (Dellacherie and Meyer (1978), Theorem IV.67). It suffices to show that X(T, ω, y) = X(T, ω, y T ) because we then prove the claim with ψ˜ = X(T ). For this, first consider X(t, ω, y) = 1(s,u] (t)1A (ω)1B (y) for u > s ≥ τ , A ∈ Fs and B ∈ Cs . Then the above claim is true because 1B (y) = 1B (y s ) and so on {s < T (ω) ≤ u}, 1B (y) = 1B (y T (ω)). The aforementioned standard bootstrapping now gives the claim for any (Fˆt )-predictable X and so completes the proof of (a). (b) First consider g(s, ω, y) = φ(ω, y)1(u,v](s) where φ ∈ bFˆu , τ ≤ u < v. Then P-a.s. for t ≥ u, t gs (ω, y)dsKt(dy) = Kt (φ)(t ∧ v − t ∧ u) τ

t φ(ω, y)(s ∧ v − s ∧ u)dM (s, y)

= u

t 1(u < s ≤ v)Ks (φ)ds (by (a) and integration

+ τ

by parts)   t s t  gr (ω, y)dr  dM (s, y) + Ks (gs )ds. = τ

τ

τ

If t < u, the above equality holds because both sides are zero. The result therefore holds for linear combinations of the above functions, i.e., for (Fˆt )-simple g. Passing to the bounded pointwise closure we obtain the result for all (Fˆt )-predictable and bounded g. For g as in (b), we first get the result for t ≤ N by considering gs∧N , and then for all t by letting N → ∞. ˆ → R is (Fˆt∗ )-predictable and bounded, and Remarks V.2.5. (a) If g : [τ, ∞) × Ω ˆ µ is a σ-finite measure on Ω, then there are bounded (Fˆt )-predictable processes g1 ≤ g ≤ g2 such that g1 (t) = g2 (t) ∀t ≥ τ µ-a.e. This may be proved by starting n with a simple g (i.e. g(t, ω, y) = φi (ω, y)1(ui,ui+1 ] (t) + φ0 (ω, y)1{u0=t} , where i=1

τ = u0 < . . . < un+1 ≤ ∞, φi ∈ bFˆu∗i ) and using a monotone class theorem as on p. 134 of Dellacherie and Meyer (1978).

Superprocesses (b) If we take

289

 ∞ 1A (ω, y)Ks(dy)ds µ(A) = P  τ

in the above, then the right side of (V.2.12) is the same for g1 , g2 and g. Here we have used the obvious extension of the stochastic integral with respect to M to (Fˆt∗ )-predictable integrands. It follows from Proposition V.2.4 (b) that the left-hand side is the same for g1 and g2 . By monotonicity it is the same for g and so (V.2.12) holds for (Fˆt∗ )-predictable, bounded g. A straightforward truncation argument then gives it for (Fˆt∗ )-predictable g satisfying t

s (V.2.13) τ

|gr |dr

2

 + |gs | Ks (dy)ds < ∞ ∀t > 0

P-a.s.

τ

(c) In Proposition V.2.4 (a), if T is predictable, ψ ∈ bFˆT∗ − (i.e. (Fˆ·∗ )T − ), g(t, ω, y) = ψ(ω, y)1[T,∞)(t), (so g is (Fˆt∗ )-predictable) and we take   ∞ 1A (ω, y)Ks(dy)ds µ(A) = P  1A (ω, y)KT (dy) + T

in (a), then ψi = gi (T, ω, y) ∈ bFˆT − (gi as in (a)) and the right side of (V.2.11) is unchanged if ψ is replaced by ψi . As above, the inequality ψ1 ≤ ψ ≤ ψ2 shows the same is true of the left side. Therefore (V.2.11) remains valid if T is predictable and ψ ∈ bFˆT∗ − . ˆ → R is bounded and (Fˆ ∗ )-predictable, then Kt (φt ) is (Ft )(d) If φ : [τ, ∞) × Ω t predictable. To see this, first note that this is clear from (c) (with T = u) if ∗ . A monotone class φ(t, ω, y) = ψ(ω, y)1(u,v](t) for some τ ≤ u < v and ψ ∈ bFˆu− argument now completes the proof (see Theorem IV.67 in Dellacherie and Meyer (1978)). Here is the extension of (HM P )τ,Kτ we mentioned earlier. Proposition V.2.6. Assume φ : [τ, ∞) × C → R is a (Ct )t≥τ -predictable map for ¯ such that which there is a (Ct )t≥τ -predictable map, A¯τ,m φ = Aφ,  t  2 ¯ Aφ(s) ds < ∞ ∀ t > τ (i) Pτ,m τ t ¯ Aφ(s, y)ds, t ≥ τ is an L2 (Ct )t≥τ -martingale under Pτ,m . (ii) n(t, y) = φ(t, y) − τ

Then

t

φ(s, y)dM (s, y) +

Kt (φt ) = Kτ (φτ ) + τ

τ

t

¯ s )ds ∀t ≥ τ a.s. Ks (Aφ

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The stochastic integralis a continuous L2 martingale, Kτ (φτ ) is square integrable, 
t ˜ s |)ds < ∞ for all t ≥ τ . and P τ Ks (|Aφ Proof. Note that (i), Cauchy-Schwarz and the superprocess property (V.2.2) show ¯ that (V.2.13) holds for g = Aφ–in fact the expression there is integrable. Therefore we may use Remark V.2.5 (b) and Lemma V.2.3 to see that P-a.s. for all t ≥ τ ,   t ¯ Aφ(s) ds Kt (φt ) = Kt (nt ) + Kt τ t s t  ¯ r (y)dr dM (s, y) Aφ = Kτ (nτ ) + n(s, y)dM (s, y) + τ τ τ t ¯ s )ds Ks (Aφ + τ t t ¯ s )ds. Ks (Aφ φ(s, y)dM (s, y) + = Kτ (φτ ) + τ

τ

Lemma V.2.3 shows that Kτ (φτ ) = Kτ (nτ ) is square integrable and (V.2.2), (i) and (ii) show that  t  t 2 Ks (φs )ds = Pτ,m (φ(s, B)2 )ds < ∞. P τ

τ

This shows the stochastic integral is an L2 martingale and a similar argument shows that the drift term has integrable variation on bounded intervals. Notation. Let D(A¯τ,m ) denote the class of φ considered in the above Proposition. ˆ ˆ ⊂ D(A¯τ,m ) and A¯τ,m is an extension of Aˆ in that for any φ ∈ D(A), Clearly D(A) t t ˆ A¯τ,m φ(s, y)ds = Aφ(s, y)ds ∀t ≥ τ Pτ,m − a.s. τ

τ

Exercise V.2.1. Assume τ = 0 and K is an (Ft )-historical Brownian motion starting at (0, ν)–note we can treat K0 as a finite measure on Rd . Let Z0 : Rd → Rd be a Borel map, B ∈ B(Rd ) and define Z˜0 : C → C by Z˜0 (y)(t) = y(t) − y(0) + Z0 (y(0)). ˜ y) = φ(t, Z˜0 (y))1B (y(0)) ∈ D(A¯0,m ) and ˆ show that φ(t, (a) If φ ∈ D(A), ˜ y) = Aφ(t, ˆ A¯0,m φ(t, Z˜0 (y))1B (y(0)). (b) Define Kt (F ) = Kt ({y : Z˜0 (y) ∈ F, y(0) ∈ B}) for F ∈ C, and let ν  be the law of K0 = K0 (Z0−1 (A) ∩ B) for A ∈ B(Rd ). Show that K  is an (Ft )-historical Brownian motion starting at (0, ν  ) and therefore has law Q0,ν  . Hint. Use (a) to show that K  satisfies (HM P )0,Kτ . Corollary V.2.7. Let T ∈ Tb and assume n(t), t ≥ τ is a (Ct )-predictable square integrable martingale under Pτ,m . Then n(t ∧ T ), t ≥ τ is an (Fˆt )-martingale under ˆT . P

Superprocesses

291

Proof. Let s ≥ τ , A ∈ Fs and B ∈ Cs . Define φ(t, y) = (n(t, y) − n(s, y))1B (y)1(t ≥ s). Then φ ∈ D(A¯τ,m ) and A¯τ,m φ = 0. Therefore Kt (φt ) is an (Ft )-martingale by Proposition V.2.6 and so ˆ T ((n(T ) − n(s ∧ T ))1A (ω)1B (y)) = P(KT (φT )1A∩{T >s} )m(1)−1 P = P(Ks (φs )1A∩{T >s} )m(1)−1 = 0, the last because φs = 0. ∞ Example V.2.8. Recall CK (Rk ) is the set of infinitely differentiable functions on k R with compact support and define

yt ), Df d = {φ : R+ × C → R : φ(t, y) = ψ(yt1 ∧t , . . . , ytn ∧t ) ≡ ψ(¯ ∞ 0 ≤ t1 ≤ . . . ≤ tn , ψ ∈ CK (Rnd )}.

If φ is as above, let φi (t, y) =

n 

1(t < tk )ψ(k−1)d+i (¯ yt ) 1 ≤ i ≤ d,

k=1 n n  

φij (t, y) =

1(t < tk ∧ t )ψ(k−1)d+i,(−1)+j (¯ yt ) 1 ≤ i, j ≤ d,

k=1 =1

¯ ∇φ(t, y) = (φ1 (t, y), . . . , φd (t, y)), and ∆φ(t, y) =

d 

φii (t, y).

i=1 ¯ Itˆ o’s Lemma shows that for any m ∈ MF (C), Df d ⊂ D(A¯τ,m ) and Aτ,m φ = ∆φ 2 for ∞ 2 φ ∈ Df d . In fact this remains true if CK is replaced with Cb in the above. Note ˆ because ∆φ(t, ¯ y) may be discontinuous in t. that Df d is not contained in D(A)

Theorem V.2.9. Let m ∈ MFτ (C). An (Ft )t≥τ -adapted process (Kt )t≥τ with sample paths in ΩH [τ, ∞) is an (Ft )-historical Brownian motion starting at (τ, m) iff for every φ ∈ Df d , t ¯  ∆φs Mt (φ) = Kt (φt ) − m(φτ ) − Ks ds 2 τ is a continuous (Ft )-local martingale such that Mτ (φ) = 0 and t Ks (φ2s )ds for all t ≥ τ a.s. M (φ)t = τ

Proof. The previous Example and Proposition V.2.6 show that an (Ft )-historical Brownian motion does satisfy the above martingale problem. The proof of the converse uses a generalization of the stochastic calculus developed in the next section for historical Brownian motion. It is proved in Theorem 1.3 of Perkins (1995)–see also Section 12.3.3 of Dawson (1993) for a different approach to a slightly different result. We will not use the uniqueness here although it plays

292

Superprocesses

an central role in the historical martingale problem treated in Perkins (1995) and discussed in Section 5 below. 3. Stochastic Integration on Brownian Trees Consider the question of defining the stochastic integral appearing in (SE)(a). ˆ under which y is a Brownian motion. An We first need a probability measure on Ω infinite family of such probabilities is given below. We continue to work with the (Ft )-historical Brownian motion, Kt , on (Ω, F , (Ft)t≥τ , P) starting at (τ, ν) but now set τ = 0 for convenience and so may view K0 and its mean measure, m, as finite measures on Rd . ¯  = (Ω , G, Gt , Q) be a filtered space and T be a (Gt )-stopping Definition. Let Ω ¯  is a (Gt )-Brownian motion time. An Rd -valued (Gt )-adapted process, Bt , on Ω stopped at T iff for 1 ≤ i, j ≤ d, Bti − B0i and Bti Btj − δij (t ∧ T ) are continuous (Gt )-martingales. ˆ T , y is a Brownian motion If T is a constant time, (V.2.2) shows that under P stopped at T . The next result extends this to stopping times. ˆ T , y is a (Fˆt )-Brownian motion stopped Proposition V.3.1. If T ∈ Tb , then under P at T . Proof. Apply Corollary V.2.7 with n(t, y) = yti − y0i and n(t, y) = yti ytj − y0i y0j − δij t. i ˆ T -a.s. This gives the result because yti = yt∧T P ˆ → Rn×d is (Fˆ ∗ )-predictable and Notation. f ∈ D(n, d) iff f : R+ × Ω t t f (s, ω, y)2ds < ∞ Kt − a.a. y ∀t ≥ 0 P − a.s. 0

ˆ → E, we say X = Y K-a.e. iff Definition. If X, Y : R+ × Ω X(s, ω, y) = Y (s, ω, y) for all s ≤ t for Kt -a.a. y for all t ≥ 0 P-a.s. If E is a metric space we say X is continuous K-a.e. iff s → X(s, ω, y) is continuous on [0, t] for Kt -a.a. y for all t ≥ 0 P-a.s.
T ˆ T -a.s. Therefore If T ∈ Tb and f ∈ D(n, d), then 0 f (s, ω, y)2ds < ∞ P
t
t ˆ T − f (s, ω, y)dys is uniquely the classical stochastic integral 0 f (s, ω, y)dys ≡ P 0 ˆ defined up to PT -null sets. The next result shows one can uniquely define a single process which represents these stochastic integrals for all T simultaneously. Proposition V.3.2. (a) If f ∈ D(n, d), there is an Rn -valued (Fˆt )-predictable process I(f, t, ω, y) such that t ˆ T − a.s. for all T ∈ Tb . f (s, ω, y)dy(s) ∀t ≥ 0 P

ˆT − (V.3.1) I(f, t ∧ T, ω, y) = P 0

(b) If I  (f ) is an(Fˆt∗ )-predictable process satisfying (V.3.1), then I(f, s, ω, y) = I  (f, s, ω, y) K − a.e.

Superprocesses

293

(c) I(f ) is continuous K-a.e. (d) (Dominated Convergence) For any N > 0, if fk , f ∈ D(n, d) satisfy    N 2 fk (s) − f (s) ds > ε = 0 ∀ε > 0, lim P KN

k→∞

0

then



 lim P sup Kt sup I(fk , s, ω, y) − I(f, s, ω, y) > ε

k→∞

t≤N

s≤t

= 0 ∀ε > 0.

(e) For any S ∈ Tb if fk , f ∈ D(n, d) satisfy   S  fk (s) − f (s)2 ds = 0, (V.3.2) lim P KS k→∞

then

0 P

sup Kt (sup I(fk , s) − I(f, s)2)→0 as k → ∞. t≤S

s≤t

Proof. To avoid factors of m(1)−1 we will assume m(1) = 1 throughout. (b) Let J(t, ω) = sup I(f, s, ω, y) − I  (f, s, ω, y) ∧ 1 Kt (dy). s≤t

Assume for the moment that J is (Ft )-predictable, and let T be a bounded (Ft )predictable stopping time. Then  ˆ P (J(T, ω)) = PT sup I(f, s) − I  (f, s) ∧ 1 = 0, s≤T


s ˆ T , I(f, s ∧ T ) and I  (f, s ∧ T ) are both versions of P ˆ T − f (s)dy(s). because under P 0

By the Section Theorem we see that J(t, ω) = 0 ∀t ≥ 0 P-a.s., as required. To prove J is (Ft )-predictable, let φ(t, ω, y) be the integrand in the definition of J. The projection of a B × Fˆt∗ -measurable set onto Ω is Fˆt∗ -measurable (Theorem III.13 of Dellacherie and Meyer (1978)) and so φ(t) is (Fˆt∗ )-adapted. Therefore φ(t−) is (Fˆt∗ )-predictable (being left-continuous) and hence so is φ(t) = φ(t−) ∨ (I(f, t, ω, y) − I  (f, t, ω, y) ∧ 1) . Remark V.2.5 (d) now shows that J(t) = Kt (φ(t)) is (Ft )-predictable. (a), (c) For simplicity set d = n = 1 in the rest of this proof (this only affects a few n fi (ω, y)1(ui,ui+1 ] (s), where fi ∈ bFˆui constants in what follows). If f (s, ω, y) = i=1

and 0 = u0 < . . . < un+1 ≤ ∞, (call f (Fˆt )-simple), then define I(f, t, ω, y) =

n  i=0

fi (ω, y) (y(t ∧ ui+1 ) − y(t ∧ ui )) .

294

Superprocesses

This clearly satisfies the conclusions of (a) and (c). Let f ∈ D(1, 1). As in the usual construction of the Itˆ o integral we may choose a sequence of simple functions {fk } so that  k  1 ˆ k  |f (s) − fk (s)|2 ds ≥ 2−3k  ≤ 2−k . (V.3.3) P 4 0

Define

 I(f, t, ω, y) =

limk→∞ I(fk , t, ω, y) 0

if it exists . otherwise

Clearly I(f ) is (Fˆt )-predictable. Fix a bounded (Ft )-stopping time T and let m,  ≥ n ≥ T . Use the fact that ˆ T -Itˆ o integral and standard properties of the I(fm , t ∧ T, ω, y) is a version of the P latter to see that  ˆ PT sup |I(fm , t, ω, y) − I(f , t, ω, y)| > 2−n t≤T

 T  2 −3n ˆ T  |fm (s) − f (s)| ds ≥ 2  + 2−n ≤P 0

 T  2 = P KT  |fm (s) − f (s)| ds > 2−3n  + 2−n 

(V.3.4)

0



 T  2 = P Kn  |fm (s) − f (s)| ds > 2−3n  + 2−n 0

(by Proposition V.2.4(a))  n  2 ˆ n  |fm (s) − f (s)| ds > 2−3n  + 2−n ≤ 3 · 2−n , ≤P 0

the last by (V.3.3) and an elementary inequality. This shows both that sup |I(fm , t, ω, y) − I(f, t, ω, y)| → 0

(V.3.5)

t≤T

ˆ T -a.s. as m → ∞ P

and t t       sup PˆT − fm (s, ω, y)dy(s) − PˆT − f (s, ω, y)dy(s)  → 0 t≤T

0

as m → ∞

0

It follows that t ˆ T -a.s. f (s, ω, y)dy(s) = I(f, t ∧ T, ω, y) ∀t ≥ 0 P

ˆT − P 0

ˆ T -a.s. P

Superprocesses

295

ˆ T -a.s. This gives (a). because the left side is constant on t ≥ T P Set m = n and let  → ∞ in (V.3.4), and use (V.3.5) to conclude that  ˆ T sup |I(fm , s, ω, y) − I(f, s, ω, y)| > 2−m ≤ 3 · 2−n , sup P T ∈Tb ,T ≤m

that is sup

T ∈Tb ,T ≤m

s≤T

   P KT sup |I(fm , s, ω, y) − I(f, s, ω, y)| > 2−m s≤T

≤ 3 · 2−m .

˜ ω) = Kt (sup |I(fm , s) −I(f, s)| > 2−m ) An application of the Section Theorem (J(t, s≤t

is (Ft )-predictable, as in (b)) gives    P sup Kt sup |I(fm , s, ω, y) − I(f, s, ω, y)| > 2−m t≤m

s≤t

≤ 3 · 2−m .

Two successive applications of Borel-Cantelli show that (V.3.6)

lim sup |I(fm , s, ω, y) − I(f, s, ω, y)| = 0

m→∞ s≤t

Kt -a.a. y ∀t ≥ 0

a.s.

This certainly implies (c). (d) Fix N > 0 and assume {fk }, f satisfy the hypotheses of (d). Argue exactly as in (V.3.4) with ε > 0 in place of 2−n , but now use Remark V.2.5 (c) in place of Proposition V.2.4 (a) and take the sup over (Ft )-predictable times to conclude  ˆ T sup |I(fk , t, ω, y) − I(f, t, ω, y)| > ε sup P t≤T T ≤N,T predictable N  ˆ N  |fk − f (s, ω, y)|2 ds > ε3  + ε. ≤P 0

The first term on the right-hand side approaches 0 as k → ∞. As ε is arbitrary, the same is true for the left-hand side. As in (b),  (t, ω) → Kt sup |I(fk , s, ω, y) − I(f, s, ω, y)| > ε s≤t

is (Ft )-predictable and the Section Theorem implies  P sup Kt sup |I(fk , s, ω, y) − I(f, s, ω, y)| > ε −→ 0 as t≤N

s≤t

k → ∞ ∀ε > 0.

The random variables on the left are all bounded by sup Kt (1) ∈ L1 and so also t≤N

converge in L1 by Dominated Convergence for all ε > 0. This is the required conclusion in (d).

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Superprocesses

(e) Let S, T ∈ Tb with T ≤ S. Doob’s maximal L2 inequality shows that   T    2 ˆ ) ≤ cPT P KT sup (I(f, s) − I(fk , s)) (f (s) − fk (s))2 ds s≤T

0

  = cP KS

T

(f (s) − fk (s))2 ds

0

  ≤ cP KS

S

(f (s) − fk (s))2 ds



 .

0

Remark V.2.5 (c) was used in the second line. Therefore (V.3.2) implies that    (V.3.7) lim sup P KT sup (I(f, s) − I(fk , s))2 = 0. k→∞ T ≤S,T ∈Tb

s≤T

As in the proof of (b), (t, ω) → Kt (sups≤t (I(f, s) − I(fk , s))2 ) is (Ft )-predictable. A simple application of the Section Theorem shows that     P sup Kt sup(I(f, s) − I(fk , s))2 > ε t≤S

=

s≤t

sup

T ≤S,T ∈Tb

    P KT sup (I(f, s) − I(fk , s))2 > ε , s≤T

which approaches zero as k → ∞ by (V.3.7). ˆ ˆ∗ Lemma Let + × Ω → R+ be 
g : R  (Ft )-predictable and S ∈ Tb .  V.3.3.
S S (a) P KS 0 gs ds = P 0 Ks (gs )ds .    
 
t S (b) P supt≤S Kt 0 gs ds > ε ≤ ε−1 P 0 Ks (gs )ds for all ε > 0. Proof. (a) By Monotone Convergence it suffices to consider g bounded. This case is then immediate from Remark V.2.5(b).  
t (b) From Remark V.2.5 (d) we see that Kt 0 gs ds ≤ ∞ is (Ft )-predictable. By the Section Theorem,     T     t gs ds > ε = sup P KT gs ds > ε P sup Kt t≤S

T ≤S,T ∈Tb

0

≤

sup

T ≤S,T ∈Tb

= ε−1 P



0

  −1 ε P KT 

S

Ks (gs )ds



T

gs ds 0

(by (a)).

0

ˆ F) ˆ and Notation. If X(t) = (X1 (t), . . . , Xn (t)) is an Rn -valued process on (Ω, µ ∈ MF (Rd ), let µ(Xt ) = (µ(X1 (t)), . . . , µ(Xn (t))) and t t   t X(s)dM (s, y) = X1 (s)dM (s, y), . . . , Xn (s)dM (s, y) , 0

0

0

Superprocesses

297


t whenever these integrals are defined. We also write 0 f (s, ω, y)dy(s) for I(f, t, ω, y) when f ∈ D(n, d) and point out that dependence on K is suppressed in either notation. Proposition V.3.4. If f ∈ D(n, d) is bounded, then t I(f, s)dM (s, y) ∀t ≥ 0 (V.3.8) Kt (I(f, t)) =

P − a.s.,

0

and the above is a continuous L2 (Ft )-martingale. Proof. To simplify the notation take n = d = 1 and m(1) = 1. Assume first that (V.3.9)

f (s, ω, y) = φ1 (ω)φ2 (y)1(u < s ≤ v),

φ1 ∈ bFu , φ2 ∈ bCu , 0 ≤ u < v.

If n(t, y) = φ2 (y)(y(t ∧ v) − y(t ∧ u)), then (n(t), Ct ) is an L2 -martingale under P0,m and I(f, t) = φ1 (ω)n(t, y). Lemma V.2.3 shows that P-a.s. for all t ≥ 0, Kt (I(f, t)) = φ1 (ω)Kt(nt ) t n(s, y)dM (s, y) = φ1 (ω) 0 t I(f, s)dM (s, y). = 0

In the last line we can take φ1 through the stochastic integral since φ1 ∈ bFu and n(s) vanishes for s ≤ u. Suppose (V.3.8) holds for a sequence of (Fˆt )-predictable processes fk and f is an (Fˆt∗ )-predictable process such that supk fk ∞ ∨ f ∞ < ∞ and  N  (fk (s) − f (s))2 dsKN (dy) = 0 ∀N ∈ N. (V.3.10) lim P k→∞

0

ˆ s , I(f ) We claim (V.3.8) also holds for f . If N ∈ N, then use the fact that under P is an ordinary Itˆ o integral to conclude  N  N ˆ s ((I(f, s) − I(fk , s))2 )ds P (I(f, s) − I(fk , s))2 Ks (dy)ds = P 0

0



N

= 0



N

ˆs P





s 0

P KN

= 0

 (f (r) − fk (r))2 dr ds



s

(f (r) − fk (r))2 dr

 ds,

0

where Remark V.2.5 (c) is used in the last line. This approaches zero as k → ∞ by
t
(V.3.10). Therefore 0 I(f, s)dM (s, y) is a continuous L2 martingale and 2  t 2 2  2 I(fk , s) − I(f, s)dM (s, y)2 = 0 ∀N ∈ N. (V.3.11) lim 2sup  k→∞ t≤N

0

2

298

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Proposition V.3.2 (e) with S = N and (V.3.10) imply P

sup Kt (|I(fk , t) − I(f, t)|)→0 as k → ∞

(V.3.12)

∀N ∈ N.

t≤N

Now let k → ∞ in (V.3.8) (for fk ) to see that it also holds for f . We may now pass to the bounded pointwise closure of the linear span of the class of f satisfying (V.3.9) to see that the result holds for all bounded (Fˆt )-predictable f . If f is bounded and (Fˆt∗ )-predictable, there is a bounded (Fˆt )-predictable f˜ so that   N (f (s) − f˜(s))2 dsKN (dy) = 0 ∀N ∈ N P 0

(see Remark V.2.5 (a)) and so the the result holds for f by taking fk = f˜ in the above limiting argument. Theorem V.3.5 (Itˆ o’s Lemma). Let Z0 be Rn -valued and Fˆ0∗ -measurable, f ∈ D(n, d), g be an Rn -valued (Fˆt∗ )-predictable process and ψ ∈ Cb1,2 (R+ × Rn ). Assume t Ks (fs 2 + gs )ds < ∞ ∀t > 0 a.s., (V.3.13) 0

and let



(V.3.14)



t

Zt (ω, y) = Z0 (ω, y) +

f (s, ω, y)dys + 0

t

g(s, ω, y)ds. 0

If ∇ψ and ψij denote the gradient and second order partials in the spatial variables, then t ψ(t, Zt)Kt (dy) = ψ(0, Z0 )dK0 (y) + ψ(s, Zs)dM (s, y) (V.3.15)

t

+

Ks 0

 ∂ψ ∂s

0

 1  ψij (s, Zs )(f f ∗ )ij (s) ds. 2 i=1 j=1 n

(s, Zs ) + ∇ψ(s, Zs) · gs +

n

The second term on the right is an L2 martingale and the last term
on the right has paths with finite variation on compact intervals a.s. In particular ψ(t, Zt )Kt (dy) is a continuous (Ft )-semimartingale. Proof. Assume first that f  and g are bounded. Let T ∈ Tb and Zt = (t, Zt ). ˆ T -a.s. Itˆo’s Lemma shows that P t∧T ˆ ∇ψ(Zs )f (s) · dy(s) ψ(Zt∧T ) =ψ(Z0 ) + PT − + 0

0 t∧T

∂ψ 1  ψij (Zs )(f f ∗ )ij (s)ds ∀t ≥ 0. (Zs ) + ∇ψ(Zs ) · g(s) + ∂s 2 n

n

i=1 j=1

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299

Let ˜b(s) denote the integrand in the last term. This shows that t ˜b(s)ds ˜ I(t) = ψ(Zt ) − ψ(Z0 ) − 0

(Fˆt∗ )-predictable

process satisfying (V.3.1) with ∇ψ(Zs )f (s) in place of f (s). is a Proposition V.3.2 therefore implies that t ˜b(s)ds K − a.e. (V.3.16) ψ(Zt ) = ψ(Z0 ) + I(∇ψ(Z)f, t) + 0

(FˆT∗ )-predictable

we may apply PropoSince ˜b and ∇ψ(Z)f  are bounded and sition V.3.4 and Remarks V.2.5 to see that P-a.s. for all t ≥ 0, ψ(Zt )Kt (dy) t t = ψ(Z0 )K0 (dy) + ψ(Z0 )dM (s, y) + I(∇ψ(Z)f, s)dM (s, y) 0 0 t s  t ˜b(r)dr dM (s, y) + + Ks (˜b(s))ds 0 0 0 t t Ks (˜b(s))ds. ψ(Zs )dM (s, y) + = ψ(Z0 )K0 (dy) + 0

0

In the last line we used (V.3.16) and the fact that this implies the stochastic integrals of both sides of (V.3.16) with respect to M coincide. This completes the proof in this case. Assume now that f, g satisfy (V.3.13). By truncating we may choose bounded (Ft∗ )-predictable f k , g k such that f k → f and g k → g pointwise, f k  ≤ f  and g k  ≤ g pointwise, and therefore t Ks (fsk − fs 2 + gsk − gs )ds = 0 ∀t ≥ 0 a.s. (V.3.17) lim k→∞

0

By (V.3.13) we may choose Sn ∈ Tb satisfying Sn ↑ ∞ a.s. and Sn Ks (fs 2 + gs )ds ≤ n a.s. 0 k

k k Define Z as in (V.3.14) but with 
(f , g ) in place of (f, g). Note that Lemma t V.3.3(b) shows that supt≤Sn Kt 0 gs ds < ∞ for all n a.s. and so Zt (ω, y) is well-defined Kt -a.a. y for all t ≥ 0 a.s. The same result shows that   t P gsk − gs ds →0 as k → ∞ ∀n. (V.3.18) sup Kt t≤Sn

0

A similar application of Proposition V.3.2 (e) and Lemma V.3.3(a) and gives (V.3.19)

P

sup Kt (sup I(f k , s) − I(f, s)2)→0 as k → ∞ ∀n.

t≤Sn

s≤t

300

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(V.3.18) and (V.3.19) imply (V.3.20)

P

sup Kt (sup Z k (s) − Z(s))→0 as k → ∞ ∀T > 0. t≤T

s≤t

We have already verified (V.3.15) with (Z k , f k , g k ) in place of (Z, f, g). The boundedness of ψ and its derivatives, together with (V.3.17) and (V.3.20), allow us to let k → ∞ in this equation and use Dominated Convergence to derive the required result. (Clearly (V.3.13) implies the “drift” term has bounded variation on bounded time intervals a.s.) Corollary V.3.6. If f , g, and Z are as in Theorem V.3.5, then Xt (A) = 1(Zt (ω, y) ∈ A)Kt (dy) process. defines an a.s. continuous (Ft )-predictable MF (Rn )-valued
Proof. Take ψ(x) = eiu·x , u ∈ Qn to see that eiu·x dXt (x) is continuous in t ∈ [0, ∞) for all u ∈ Qn a.s. L´evy’s Continuity Theorem completes the proof of continuity. Remark V.2.5 (d) shows that X is (Ft )-predictable. 4. Pathwise Existence and Uniqueness As in the last Section we assume (Kt , t ≥ 0) is an (Ft )-historical Brownian mo¯ = (Ω, F , Ft, P) starting at (0, ν) with E(K0 (·)) = m(·) ∈ MF (Rd ). Theretion on Ω Recall that σ, b satisfy (Lip) and fore K has law Q0,ν on (ΩX , FX ).
t Z0 : Rd → Rd is a Borel map. If 0 σ(Xs , Zs )dy(s) is the stochastic integral introduced in Proposition V.3.1, here is the precise interpretation of (SE): t t (SE)Z0 ,K (a) Z(t, ω, y) = Z0 (y0 ) + σ(Xs , Zs )dy(s) + b(Xs , Zs )ds K − a.e. 0 0 (b) Xt (ω)(A) = 1(Zt (ω, y) ∈ A)Kt (ω)(dy) ∀A ∈ B(Rd ) ∀t ≥ 0 a.s. (X, Z) is a solution of (SE)Z0 ,K iff Z is a (Fˆt∗ )-predictable Rd -valued process, and ¯t X is an (Ft )-predictable MF (Rd )-valued process such that (SE)Z0 ,K holds. Let H denote the usual enlargement of F H [0, t+] with Q0,ν -null sets. Theorem V.4.1. (a) There is a pathwise unique solution (X, Z) to (SE)Z0 ,K . More precisely X is unique up to P-evanescent sets and Z is unique K-a.e. Moreover t → Xt is a.s. continuous in t. ˆ¯ ∗ )-predictable maps, X ˜ : R+ ×ΩH → MF (Rd ) ¯ t )-predictable and (H (b) There are (H t d ˆ ˜ and Z : R+ ×ΩH → R , respectively, which depend only on (Z0 , ν), and are such that ˜ K(ω)), Z(t, ˜ K(ω), y)) defines the unique solution that (X(t, ω), Z(t, ω, y)) = (X(t, of (SE)Z0 ,K . (c) There is a continuous map X0 → PX0 from MF (Rd ) to M1 (ΩX ), such that if ¯ then (X, Z) is a solution of (SE)Z0 ,K on some filtered space Ω, (V.4.1) P(X ∈ ·) = PX0 (ω) dP(ω).

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301

(d) If T is an a.s. finite (Ft )-stopping time, then P(X(T + ·) ∈ A|FT )(ω) = PXT (ω) (A) P − a.s. for all A ∈ FX . For the uniqueness and continuity of PX0 we will need to prove the following stability result. Recall that d is the Vasershtein metric on MF (Rd ). Theorem V.4.2. Let K 1 ≤ K 2 ≡ K be (Ft )-historical Brownian motions with E(K0i (·)) = mi (·) ∈ MF (Rd ), and let Z0i : Rd → Rd be Borel maps, i = 1, 2. Set TN = inf{t : Kt (1) ≥ N } ∧ N . There are universal constants {cN , N ∈ N} so that if (X i , Z i ) is the unique solution of (SE)Z0i ,K i then P



TN

0



sup d(X (u), X (u)) ds ≤ cN 1

2

2



u≤s

 Z01 − Z02 2 ∧ 1dm2 + m2 (1) − m1 (1) .

Here are the metric spaces we will use in our fixed point argument: S1 = {X : R+ × Ω → MF (Rd ) : X is (Ft ) − predictable and a.s. continuous , Xt (1) ≤ N ∀t < TN ∀N ∈ N}, d ∗ ˆ → R : Z is (Fˆ ) − predictable and continuous K-a.e.}, S2 = {Z : R+ × Ω t

S = S1 × S2 . Processes in S1 are identified if they agree at all times except perhaps for a P-null set and processes in S2 are identified if they agree K-a.e. If T ∈ Tb , θ > 0 and Z1 , Z2 ∈ S2 , let 1/2  T ˆT (V.4.2) dT,θ (Z1 , Z2 ) = P (sup Z1 (u) − Z2 (u)2 ∧ 1)e−θs ds ≤ θ −1/2 , u≤s

0

ˆ T is the normalized Campbell measure associated with KT . If where, as usual, P θ¯ = (θN , N ≥ 1) is a positive sequence satisfying ∞ 

(V.4.3)

−1/2

N θN

< ∞,

1 ¯

¯

define metrics di = dθi on Si and d0 = dθ0 on S by d1 (X1 , X2 ) = d2 (Z1 , Z2 ) =

∞   P N=1 ∞  N=1

TN

0

sup

sup d(X1 (u), X2 (u))2 e−θN s ds

1/2 ,

u≤s

T ≤TN ,T ∈Tb

dT,θN (Z1 , Z2 ) < ∞ (by (V.4.2)),

d0 ((X1 , Z1 ), (X2 , Z2 )) = d1 (X1 , X2 ) + d2 (Z1 , Z2 ).

302

Superprocesses

Note that if u < TN , then d(X1 (u), X2 (u)) ≤ X1 (u)(1) + X2 (u)(1) ≤ 2N and so by (V.4.3),  −1/2 2N θN < ∞. d1 (X1 , X2 ) ≤ N

Lemma V.4.3. (Si , di ), i = 1, 2 and (S, d0 ) are complete metric spaces. Proof. This is straightforward. We will only show d2 is complete. Suppose {Zn } is d2 Cauchy. Let N ∈ N. Then  T  lim sup E sup Zn (u) − Zm (u)2 ∧ 1dsKT (dy) = 0. m,n→∞ T ≤TN ,T ∈T

b

0

u≤s

An application of the Section Theorem implies that t P (sup Zn (u) − Zm (u)2 ∧ 1)dsKt (dy)→0 as m, n → ∞. sup t≤TN

0

u≤s

Since Zn − Zm is continuous K-a.e., this implies P sup Zn (s) − Zm (s)2 ∧ 1Kt (dy)→0 as m, n → ∞. sup t≤TN

s≤t

A standard argument now shows that there is a subsequence {nk } so that (V.4.4) Znk (s, ω, y) converges uniformly in s ≤ t for Kt − a.e. y for all t ≥ 0 P − a.s. Now define

 Z(s, ω, y) =

limk→∞ Znk (s, ω, y) if it exists otherwise. 0 ∈ Rd

Then Z is (Fˆt∗ )-predictable and continuous K-a.e. (by (V.4.4)). Dominated Convergence easily shows that d2 (Znk , Z) → 0 as k → ∞. Proof of Theorem 4.1(a) and Theorem 4.2. For i = 1, 2 define Φi = (Φi1 , Φi2 ) : S → S by t t i i i ˜ σ(Xs , Zs )dy(s) + b(Xs , Zs )ds (V.4.5) Φ2 (X, Z)(t) ≡ Zt (ω, y) = Z0 (y0 ) + 0

and

0

˜ i (·) = Φi1 (X, Z)(t)(·) ≡ X t

(V.4.6)

1(Z˜ti (ω, y) ∈ ·)Kti (dy).

In (V.4.5) the stochastic integral is that of Section 3 relative to K. Note that this integral also defines a version of the K 1 stochastic integral. To see this, note it is trivial for simple integrands and then one can approximate as in the construction of the K-stochastic integral to see this equivalence persists for all integrands in D(d, d) (defined with respect to K). Remark V.1.1(a) implies t (V.4.7) Ks (σ(Xs , Zs )2 + b(Xs , Zs ))ds 0

Superprocesses

303

t

Ks (1)(C(Xs (1))2 + C(Xs (1)))ds < ∞ ∀t > 0 a.s.

≤ 0

˜ is a.s. continuous. Note also Corollary V.3.6 therefore may be used to show that X ˜ that Xt (1) = Kt (1) ≤ N for t < TN and it follows that Φi takes values in S. Clearly a fixed point of Φi would be a solution of (SE)Z0i ,K i . To avoid writing factors of m(1)−1 , assume that m(1) = 1. Let T ∈ Tb satisfy ˜ i be as in (V.4.5) and (V.4.6) for some (X i , Z i ) ∈ S. Doob’s T ≤ TN . Let Z˜ i and X 2 strong L inequality and Cauchy-Schwarz imply   ˆ T sup Z˜ 1 (s) − Z˜ 2 (s)2 ∧ 1 P s≤t∧T

 1 2 2 ˆ ≤ cPT Z0 − Z0  ∧ 1 +

t∧T

σ(Xs1 , Zs1 ) − σ(Xs2 , Zs2 )2 0

+ N b(Xs1 , Zs1 ) − b(Xs2 , Zs2 )2 ds





Z01 − Z02 2 ∧ 1dm2  t∧T  ˆT (d(Xs1 , Xs2 ) + Zs1 − Zs2 )2 ∧ C(N )2 ds , + cL(N )2 N P

≤c

(V.4.8)

0

where in the last line we used Proposition V.2.4 (a), (Lip) and Remark V.1.1 (a). Note that if T > 0 then TN > 0 and so KT (1) ≤ supt≤TN Kt (1) ≤ N . If T = 0, KT (1) may be bigger than N but the integral in (V.4.8) is then zero. Therefore   ˆ T sup Z˜ 1 (s) − Z˜ 2 (s)2 ∧ 1 P s≤t∧T

≤ cN

(V.4.9)



Z01 − Z02 2 ∧ 1dm2 + P



t∧T

d(Xs1 , Xs2 )2 ∧ 1ds

0

ˆT +P





t∧T

Zs1 − Zs2 2 ∧ 1ds

 .

0

Multiply the above inequality by e−θt and integrate t over R+ to conclude that sup

T ≤TN ,T ∈Tb

≤

dT,θ (Z˜ 1 , Z˜ 2 )2

 TN  cN

(d(Xs1 , Xs2 )2 ∧ 1)2 e−θs ds Z01 − Z02 2 ∧ 1dm2 + P θ 0  T  ˆT + sup (Zs1 − Zs2 2 ∧ 1)e−θs ds . P T ≤TN ,T ∈Tb

Therefore sup

T ≤TN ,T ∈Tb

(V.4.10)

0

3 dT,θ (Z˜ 1 , Z˜ 2 ) ≤

cN  θ

Z01 − Z02 2 ∧ 1dm2

+P

 0

TN

1/2

(d(Xs1 , Xs2 )2 ∧ 1)2 e−θs ds

1/2

304

Superprocesses +

sup

T ≤TN ,T ∈Tb

 dT,θ (Z 1 , Z 2 ) .

Take θ = θN in (V.4.10), assume δ0 =

(V.4.11)

∞ 3  cN < ∞, θN

N=1

and sum the resulting inequality over N to conclude that 1/2 

 + d1 (X 1 , X 2 ) + d2 (Z 1 , Z 2 ) . (V.4.12) d2 (Z˜ 1 , Z˜ 2 ) ≤ δ0 Z01 − Z02 2 ∧ 1dm ˜ 1, X ˜ 2 ). Let φ ∈ Lip . Then Consider next a bound for d1 (X 1

 2  2  ˜ 2 (φ)|2 ≤ 2 ˜ 1 (φ) − X φ(Z˜u1 ) − φ(Z˜u2 )dKu1 + φ(Z˜u2 )d(Ku1 − Ku2 ) |X u u

 2  ≤2 Z˜u1 − Z˜u2  ∧ 2dKu1 + (Ku2 (1) − Ku1 (1))2 .


sups≤u Z˜s1 − Z˜s2  ∧ 1Ku1 (dy), then 

˜ 1, X ˜ 2 )2 ≤ 8 sup N (u)2 + sup (K 2 (1) − K 1 (1))2 . sup d(X u u u u

Therefore if N (u) = (V.4.13)

u≤t∧TN

u≤t∧TN

u≤t∧TN

Claim N is an (Ft )-submartingale. Let U ≤ V be (Ft )-stopping times. Then Remark V.2.5 (c) implies   E(N (U )) = E sup Z˜s1 − Z˜s2  ∧ 1 dKV ≤ E(N (V )). s≤U

As N (t) ≤ Kt (1) ∈ L1 , the claim follows by a standard argument. On {TN > 0} we have supt≤TN Kt (1) ≤ N . Therefore (V.4.13), Cauchy-Schwarz and Doob’s strong L2 inequality imply ˜ 1, X ˜ 2 )2 ) P(1(TN > 0) sup d(X u u u≤t∧TN

2 1 ≤ cP([N (t ∧ TN )2 + (Kt∧T (1) − Kt∧T (1))2 ]1(TN > 0)) N N   2 2 1 + N (Kt∧T (1) − Kt∧T (1)) ≤ cP N sup Z˜s1 − Z˜s2 2 ∧ 1 dKt∧T N N N s≤t∧TN

   ˆT = cN P sup Z˜s1 − Z˜s2 2 ∧ 1 + m2 (1) − m1 (1) (by Remark V.2.5 (c)) N ≤ cN



s≤t∧TN

Z01 − Z02 2 ∧ 1 dm2 + m2 (1) − m1 (1) + P ˆT +P N



 0

t∧TN

0 t∧TN

d(Xs1 , Xs2 )2 ∧ 1 ds

Zs1 − Zs2 2 ∧ 1 ds



 ,

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305

the last by (V.4.9). It follows that   TN ˜ 1, X ˜ 2 )2 e−θN t dt P sup d(X u u 0



≤

u≤t ∞ 

 ˜ u1 , X ˜ u2 )2 e−θN t dt P 1(TN > 0) sup d(X u≤t∧TN

0

cN

≤ Z01 − Z02 2 ∧ 1 dm2 + m2 (1) − m1 (1) θN    TN ˆT (d(Xs1 , Xs2 )2 ∧ 1) e−θN s ds + P +P N 0

δ0

(V.4.14)

=

∞  1

(V.4.15)

˜ 1, X ˜ 2) ≤ δ d1 (X 0

(Zs1 − Zs2 2 ∧ 1) e−θN s ds



0

Therefore if

then

TN



4

cN < ∞, θN

1/2

Z01 − Z02 2 ∧ 1 dm2

+ (m2 (1) − m1 (1))1/2  + d1 (X 1 , X 2 ) + d2 (Z 1 , Z 2 ) .

Now set Z01 = Z02 = Z0 and K 1 = K 2 = K and Φi = Φ. (V.4.12) and (V.4.15) imply d0 (Φ(X 1 , Z 1 ), Φ(X 2 , Z 2 )) ≤ (δ0 + δ0 )d0 ((X 1 , Z 1 ), (X 2 , Z 2 )). Therefore if we choose {θN } so that (V.4.11) and (V.4.14) hold with δ0 , δ0 ≤ 1/4, then Φ is a contraction on the complete metric space (S, d0 ). It therefore has a unique fixed point (X, Z) which is a solution of (SE)Z0 ,K . Conversely if (X, Z) is a solution of (SE)Z0 ,K then Xs (1) = Ks (1) and, as in (V.4.7) and the ensuing argument, we see that (X, Z) ∈ S. Therefore (X, Z) is a fixed point of Φ and so is pathwise unique. This completes the proof of Theorem V.4.1 (a). Then For Theorem 4.2, let (X i , Z i ) satisfy (SE)Z0i ,K i , i = 1, 2. i (X , Z i ) ∈ S by the above and so Φi (X i , Z i ) = (X i , Z i ). Therefore (V.4.12) and (V.4.15) imply

d0 ((X 1 , Z 1 ), (X 2 , Z 2 )) ≤(δ0 + δ0 ) d0 ((X 1 , Z 1 ), (X 2 , Z 2 ))  1/2  + (m2 (1) − m1 (1))1/2 . + Z01 − Z02 2 ∧ 1 dm2 As δ0 + δ0 ≤ 1/2 by our choice of θN , this gives d1 (X 1 , X 2 ) ≤ d0 ((X 1 , Z 1 ), (X 2 , Z 2 ))  1/2 ≤ + (m2 (1) − m1 (1))1/2 , Z01 − Z02 2 ∧ 1 dm2

.

306

Superprocesses

and hence   TN sup d(X 1 (u), X 2(u))2 ds P 0

u≤s

  ≤ eθN N d1 (X 1 , X 2 )2 ≤ 2eθN N m2 (1) − m1 (1) + Z01 − Z02 2 ∧ 1 dm2 . ˜ Z) ˜ be the unique solution of (SE)Z ,H , where Proof of Theorem 4.1(b). Let (X, 0 ¯ t , Q0,ν ). Clearly (X, ˜ Z) ˜ depends ¯ H is the canonical process on ΩH = (ΩH , FH , H only on Z0 and ν, and satisfies the required predictability conditions. We must show ¯ Let IH (f, t, H, y), f ∈ DH ˜ K(ω)), Z(t, ˜ K(ω), y)) solves (SE)Z ,K on Ω. that (X(t, 0 ¯ H and I(f, t, ω, y), f ∈ D denote the stochastic integral from Proposition V.3.2 on Ω ¯ We claim that if f ∈ DH , then continue to denote that with respect to K on Ω. f ◦ K(t, ω, y) ≡ f (t, K(ω), y) ∈ D and I(f ◦ K) = IH (f ) ◦ K

(V.4.16)

K − a.e.

The first implication is clear because K has law Q0,ν . (V.4.16) is immediate if 5¯ )-predictable simple function. A simple approximation argument using f is a (H t Proposition V.3.2 (d) then gives (V.4.16) for all f ∈ DH . It is now easy to replace ˜ ◦ K, Z˜ ◦ K) solves (SE)Z ,K on Ω. ¯ H with K(ω) in (SE)Z0 ,H and see that (X 0 For (c) we need: Lemma V.4.4. Let µ1 , µ2 ∈ MF (Rd ) satisfy µ1 (Rd ) ≤ µ2 (Rd ). There is a measure m ∈ MF (Rd ), B ∈ B(Rd ), and Borel maps gi : Rd → Rd such that (1) µ1 (·) = m(g1−1 (·) ∩ B), µ2 (·) = m(g2−1 (·)) (2) g1 − g2  ∧ 1 dm ≤ 2d(µ1 , µ2 ) (3) m(B c ) = µ2 (1) − µ1 (1) ≤ d(µ1 , µ2 ). Proof. (3) is immediate from (1). If µ1 (Rd ) = µ2 (Rd ), (1) and (2) with B = Rd and no factor of 2 in (2) is a standard “marriage lemma” (see, e.g., Szulga (1982)). Although the usual formulation has m defined on the Borel sets of (Rd )2 and gi the projection maps from (Rd )2 onto Rd , the above results follows as m and gi can be carried over to Rd through a measure isomorphism between this space and (Rd )2 . If µ1 (Rd ) < µ2 (Rd ), let µ1 = µ1 + (µ2 (Rd ) − µ1 (Rd ))δx0 , where x0 is chosen so that µ1 ({x0 }) = 0. By the above case there are m, gi satisfying (1) and (2) with µ1 in place of µ1 , B = Rd , and no factor of 2 in (2). (1) follows easily with B = g1−1 (x0 )c . For (2) note that g1 − g2  ∧ 1 dm ≤ d(µ1 , µ2 ) (by the above case) ≤ d(µ1 , µ1 ) + d(µ1 , µ2 ) = µ2 (1) − µ1 (1) + d(µ1 , µ2 ) ≤ 2d(µ1 , µ2 ).

Superprocesses

307

Proof of Theorem 4.1(c). Assume first that K0 = m is deterministic and let (X, Z) solve (SE)Z0 ,K . The law of X on ΩX , Pm,Z0 , depends only on (m, Z0 ) by (b). The next step is to show that it in fact only depends on X0 = m(Z0−1 (·)). Define Z˜0 (y)(t) = y(t) − y(0) + Z0 (y(0)) and Kt (φ) = Kt (φ ◦ Z˜0 ). Then K  is an (Ft )-historical Brownian motion starting at (0, X0 ) by Exercise V.2.1 with B = Rd . Let (X  , Z  ) be the unique solution of (SE)Z0 ,K  , where Z0 (y0 ) = y0 . Let I, I  denote the stochastic integrals on D(d, d) and D (d, d), respectively with respect to ˆ → Rd×d , set f ◦ Z˜0 (t, ω, y) = f (t, ω, Z ˜0(y)). K and K  , respectively. If f : R+ × Ω  ˜ We claim that if f ∈ D , then f ◦ Z0 ∈ D and I  (f ) ◦ Z˜0 = I(f ◦ Z˜0 ) K − a.e.

(V.4.17)

The first inclusion is trivial. To prove the equality note it is obvious if f is simple and then use Proposition V.3.2(d) to extend the equality to all f in D by approximating by simple functions as in the construction of I. If Zˆt = Zt ◦ Z˜0 , then Zˆ is (Fˆt∗ )predictable, and t Zt = y0 + I  (σ(X  , Z  ), t) + b(Xs , Zs )ds K  − a.e. 0

implies

Zˆt = Z˜0 (y)(0) + I  (σ(X  , Z  ), t) ◦ Z˜0 +

t

b(Xs , Zs ) ◦ Z˜0 ds K − a.e.

0

Now use (V.4.17) to get ˆ t) + Zˆt = Z0 (y0 ) + I(σ(X  , Z),



t

b(Xs , Zˆs )ds K − a.e.,

0

and also note that P-a.s. for all t ≥ 0, Xt (·) = 1(Zt ∈ ·)Kt (dy) = 1(Zˆt ∈ ·)Kt (dy). ˆ solves (SE)Z ,K and so X  = X P-a.s. by (a). This implies they Therefore (X  , Z) 0 have the same law and so Pm,Z0 = PX0 ,id ≡ PX0 , thus proving the claim. To show the continuity of PX0 in X0 we will use the stability of solutions to (SE) with respect to the initial conditions (Theorem V.4.2). Let X0i ∈ MF (Rd ), i = 1, 2 and choose m, B, and gi ≡ Z0i as in Lemma V.4.4 with µi = X0i . Let Kt2 = Kt be an (Ft )-historical Brownian motion starting at (0, m) and define Kt1 (A) = Kt (A ∩ {y : y(0) ∈ B}). Then Exercise V.2.1 shows that K 1 (≤ K 2 ) is an (Ft )-historical Brownian motion starting at (0, m(· ∩ B)). If (X i , Z i ) solves (SE)Z0i ,K i , then X i (0) = X0i , as the notation suggests, and so Xi has law PX i . Introduce the uniform metric 0

ρM (x1 , x2 ) = supt d(x1t , x2t ) ∧ 1 on ΩX and let dM denote the corresponding Vasershtein metric on MF (ΩX ). This imposes a stronger (i.e., uniform) topology on ΩX , and hence on MF (ΩX ), but as our processes have compact support in time the

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strengthening is illusory. If TN is as in Theorem V.4.2 and ζ = inf{t : Kt (1) = 0}, then Xti = 0 if t ≥ ζ and so |φ(X 1 ) − φ(X 2 )| dP dM (PX 1 , PX 2 ) ≤ sup 0 0 φ∈Lip1 ≤ sup d(Xt1 , Xt2 ) ∧ 1 dP t   ≤ P(TN ≤ ζ) + P 1(TN > ζ) sup d(Xt1 , Xt2 ) ∧ 1 ≤ P(TN ≤ ζ) + P



t≤TN

TN +1

TN

sup d(Xu1 , Xu2 )2 ∧ 1 du 1(TN > ζ)

1/2 ,

t≤u

the last because on {TN > ζ}, TN = N and TN+1 = N + 1. Theorem V.4.2 implies that 4     dM (PX 1 , PX 2 ) ≤ P(TN ≤ ζ) + cN+1 Z01 − Z02 2 ∧ 1 dm + m(1) − m1 (1) 0

0

≤ P(TN ≤ ζ) +

6 cN+1 3d(X01 , X02 ).

The first term approaches zero as N → ∞ and so the uniform continuity of X0 → PX0 with respect to the above metrics on M1 (ΩX ) and MF (Rd ) follows. Returning now to the general setting of (a) in which K0 and X0 may be random, we claim that if A ∈ FX , then (V.4.18)

P(X ∈ A|F0 )(ω) = PX0 (ω) (A) P − a.s.

Take expectations of both sides to complete the proof of (c). For (V.4.18), use (b) and the (Ft )-Markov property of K to see that P-a.s., (V.4.19)

˜ ˜ ∈ A). ∈ A|F0 )(ω) = Q0,K0 (ω) (X P(X ∈ A|F0 )(ω) = P(X(K)

¯ t , Q0,ν ). Let H ¯ K0 be ˜ Z) ˜ is the solution of (SE)Z ,H on (ΩH , FH , H Recall that (X, 0 the augmentation of F H [0, t+] with Q0,K0 -null sets. Claim that (V.4.20)

˜ Z) ˜ satisfies (SE)Z ,H on (ΩH , FH , H ¯ t , Q0,K ). For ν − a.a. K0 , (X, 0 0

˜ Z), ˜ t) under these various The only issue is, as usual, the interpretation of I(σ(X, measures. Let IK0 (f ), f ∈ DK0 be this integral under Q0,K0 and I(f ), f ∈ D be the integral under Q0,ν . Starting with simple functions and bootstrapping up as usual we can show for ν-a.a. K0 , f ∈ D implies f ∈ DK0 and in this case I(f, t, ω, y) = IK0 (f, t, ω, y) ∀t ≤ u for Ku − a.a. y ∀u ≥ 0 Q0,K0 − a.s. It is now a simple matter to prove (V.4.20).

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Since X0 (ω)(·) = K0 (ω)(Z0−1 (·)) a.s. by (SE)Z0 ,K , (V.4.20) and the result for deterministic initial conditions established above, imply that ˜ ∈ ·) = P Q0,K0 (ω) (X X0 (ω) (·) a.s. Use this in (V.4.19) to derive (V.4.18) and so complete the proof of (c). To establish the strong Markov property (d) we need some notation and a preliminary result. (s) Notation. If s ≥ 0 let θˆs (y)(t) = y(t + s) − y(s) and Ft = Fs+t . If Zs (ω, y) is ˆ → E, let Fˆs∗ -measurable and Z˜s (ω, y)(t) = Zs (ω, y) + θˆs (y)(t), for each φ : R+ × Ω

φ(s) (t, ω, y) = φ(t − s, ω, Z˜s (ω, y))1(t ≥ s), suppressing dependence on Zs . If B ∈ F has positive P measure, let PB (·) = P(·|B). Lemma V.4.5. Let s ≥ 0, B ∈ Fs have positive P measure and assume Zs is as above. Then (s) Kt (A) ≡ 1A (Z˜s (ω, y))Kt+s(dy), t ≥ 0 (s)

is an (Ft )-historical Brownian motion under PB . (s) Moreover PB (K0 (1)) ≤ m(1)/P(B) < ∞. The reader who has done Exercise V.2.1 should have no trouble believing this. The proof of (d) now proceeds along familiar lines. The continuity of X0 → PX0 allows us to assume T ≡ s is constant by a standard approximation of T by finitevalued stopping times. We then must show that (V.4.21)

ˆ solves (SE) ˆ (s) on There is a process Zˆ so that (Xs+· , Z) Z0 ,K (s) (Ω, F , Ft , PB ), where Zˆ0 (y) = y0 and B ∈ Fs satisfies P(B) > 0.

If B is as above and A ∈ FX then (V.4.21) and (V.4.1) show that PB (Xs+· ∈ A) = PXs (ω) (A)dPB , and this implies

P(Xs+· ∈ A|Fs )(ω) = PXs (ω) (A) a.s.,

as required. The proofs of Lemma V.4.5 and (V.4.21) are somewhat tedious and are presented in the Appendix at the end of the Chapter for completeness. It should be clear from our discussion of the martingale problem for X in Section 5 below that this is not the best way to proceed for this particular result. Now return to the martingale problem for X. Recall that a(µ, x) = σ(µ, x)σ(µ, x)∗ and Aµ φ(x) =

d  d  i=1 j=1

aij (µ, x)φij (x) +

d  i=1

bi (µ, x)φi (x),

for φ ∈ Cb2 (Rd ).

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The following exercise is a simple application of Itˆ o’s Lemma (Theorem V.3.5) and is highly recommended. Exercise V.4.1. If (X, Z) satisfies (SE)Z0 ,K show that 2 d (M P )a,b X0 For all φ ∈ Cb (R ),

t

MtX (φ) = Xt (φ) − X0 (φ) −

Xs (AXs φ)ds t X is a continuous (Ft )-martingale such that M (φ)t = Xs (φ2 )ds. 0

0

We will comment on the uniqueness of solutions to (M P )a,b X0 in the next section. Uniqueness in (SE) alone is enough to show weak convergence of the branching particle systems from Section V.1 to the solution X of (SE). Of course (Lip) continues to be in force. Theorem V.4.6. Let m ∈ MF (Rd ) and let X N be the solution of (SE)N constructed in Section V.1. If X0 = m(Z0 ∈ ·), then P(X N ∈ ·)⇒PX0 on D(R+ , MF (Rd )) as N → ∞. w

Sketch of Proof. Tightness and the fact that all limit points are supported on the space of continuous paths may be proved as in Section II.4. One way to prove that all limit points coincide is to take limits in (SE)N and show that all limit points do arise as solutions of (SE)Z0 ,K for some K. More general results are proved by Lopez (1996) using the historical martingale problem (HMP) discussed in the next section. 5. Martingale Problems and Interactions Our goal in this Section is to survey some of the recent developments and ongoing research in the martingale problem formulation of measure-valued diffusions. In order to use the uniqueness of solutions to (SE)Z0 ,K (both pathwise and in law) to show that the associated martingale problem (M P )a,b X0 is well-posed, we would have to show that it is possible to realize any solution X of (M P )a,b X0 as part of a solution (X, Z) to (SE)Z0 ,K for some historical Brownian motion K. In general it is not possible to recover K from X (see Barlow and Perkins (1994) for this result for ordinary super-Brownian motion) and so this appears to require a non-trivial enlargement of our space. Donnelly and Kurtz (1999) were able to resolve the analogous problem in the setting of their exchangeable particle representations through an elegant application of a general result of Kurtz (1998) on filtered martingale problems. Theorem V.5.1 (Donnelly-Kurtz (1999), Kurtz (1998)). If X0 ∈ MF (Rd ), (σ, b) satisfies (Lip), and a = σσ ∗ , then (M P )a,b X0 is well-posed. Discussion. Although I had originally planned to present these ideas in detail (the treatment in Section 6.5 of Donnelly and Kurtz (1999) is a bit terse), I will have to settle for a few (even more terse) remarks here. First, the ideas of Section V.4 are readily adapted to the exchangeable particle representation of Donnelly and Kurtz. Indeed it is somewhat simpler as (SE) is

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311

replaced by a countable system of (somewhat unappealing) stochastic differential equations with jumps. This pathwise uniqueness leads to the uniqueness of the martingale problem for the generator A of their exchangeable infinite particle system, (Xk (t)), and Xt (1), the total population size. The underlying population is Xt (·) Xt ∈ MF (Rd ) where for each fixed t, X is the deFinetti measure of (Xk (t)) and t (1) so (V.5.1)

Xt (φ) = Xt (1) lim (N )−1 N→∞

N 

φ(Xk (t)) a.s.

k=1

Of course X will satisfy (M P )a,b X0 and these particular solutions will be unique as the richer structure from which they are defined is unique. On the other hand, given an arbitrary solution X of (M P )a,b X0 , one can introduce ∞ ,  Xt (·) 1(Xt (1) ∈ ·) . νt (·) = E Xt (1) 1

This would be the one-dimensional marginals of ((Xk (t)), Xt (1)) if such an exchangeable system existed. Some stochastic calculus shows that νt satisfies the forward equation associated with A: t νs (Aφ) ds, φ ∈ D(A). νt (φ) = ν0 (φ) + 0

The key step is then a result of Kurtz (1998) (Theorem 2.7), earlier versions of which go back at least to Echevaria (1982). It gives conditions on A, satisfied in our setting, under which any solution of the above forward equation are the onedimensional marginals of the solution to the martingale problem associated with A. In our setting this result produces the required ((Xk (t)), Xt(1)) from which X can be recovered by (V.5.1). Here one may notice one of many simplifications we have made along the way–to obtain (V.5.1) from the martingale problem for A we need to introduce some side conditions to guarantee the fixed time exchangeability of the particle system. Hence one needs to work with a “restricted” martingale problem and a “restricted” forward equation in the above. This shows that every solution to (M P )a,b X0 arises from such an exchangeable particle system and in particular is unique in law by the first step described above. The methods of the previous section also extend easily to the historical processes underlying the solutions obtained there. Let dH be the Vasershtein metric on MF (C) associated with the metric supt yt − yt  ∧ 1 on C. Let FtH = F H [0, t+] and assume 0 H ˆ H → Rd×d , ˆb : R+ × Ω ˆ H → Rd are (F σ ˆ : R+ × Ω t )-predictable (HLip)

and for some nondecreasing function L satisfy ˆ σ (t, J, y) + ˆb(t, J, y) ≤ L(t ∨ sup Js (1)) and s≤t

ˆ σ (t, J, y) − σ ˆ (t, J  , y  ) + ˆb(t, J, y) − ˆb(t, J  , y  )

 ≤ L(t ∨ sup Js (1) ∨ sup Js (1)) sup dH (Js , Js ) + sup ys − ys  . s≤t

s≤t

s≤t

s≤t

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The historical version of (SE) is:



t

t

ˆb(s, J, Z)ds K − a.e. σ ˆ (s, J, y)dy(s) + (HSE)Z0 ,K (a) Zt (ω, y) = Z0 (y) + 0 0 (b) Jt (ω)(·) = 1(Z(ω, y)t ∈ ·)Kt (dy) ∀t ≥ 0 P − a.s. As in Section V.4, a fixed point argument shows that solutions to (HSE)Z0 ,K exist and are pathwise unique (Theorem 4.10 in Perkins (1995)). Recall the class Df d ∗ of finite dimensional cylinder functions from Example V.2.8. If aˆij = σ ˆσ ˆij , the corresponding generator is 1  AJ φ(t, y) = ˆb(t, J, y) · ∇φ(t, y) + a ˆij (t, J, y)φij (t, y), 2 d

d

φ ∈ Df d .

i=1 j=1

Assume for simplicity that K0 = m is deterministic and hence so is J0 = m(Z0 ∈ ·). It is again a simple exercise (cf. Exercise V.4.1) to show that the solution J of (HSE)Z0 ,K satisfies t ∀φ ∈ Df d Jt (φ) = J0 (φ) + AJ φ(s, y)Js (dy)ds + Mt (φ), 0 ˆ (HM P )aJˆ,0b

where Mt (φ) is a continuous (Ft )-martingale such that t M (φ)t = Js (φ2s )ds. 0

The situation in (HSE) is now symmetric in that a historical process J is constructed from a given historical Brownian motion K. If a ˆ(t, J, y) is positive definite it will be possible to reconstruct K from J so that (HSE) holds (the main steps are described below) and so we have a means of showing that any solution of (HMP) does satisfy (HSE) and hence can derive: ˆ = σ ˆσ ˆ∗ Theorem V.5.2 (Perkins (1995)). Assume (HLip), J0 ∈ MF0 (C), and a satisfies ˆ a(t, J, y)v, v > 0 ∀v ∈ Rd − {0} for Jt − a.a. y ∀t ≥ 0. ˆ

Then (HM P )aJˆ,0b is well-posed. One can use the change of measure technique in Section IV.1 to easily obˆ tain the same conclusion for (HM P )aJˆ,0b,ˆg in which AJ φ(t, y) is replaced with ˆ H → R is bounded and (Fˆ H )-predictable. AJ φ(t, y) + gˆ(t, y)φ(t, y), where gˆ : R+ × Ω t As J is intrinsically time-inhomogeneous, one should work with general starting times τ ≥ 0 and specify Jt∧τ = J 0 ∈ {H ∈ ΩH : H·∧τ = H} ≡ ΩτH . The ˆ

ˆ,b,ˆ g resulting historical martingale problem (HM P )aτ,J 0 is again well-posed and if (ˆ a, ˆb, gˆ)(t, J, y) = (˜ a, ˜b, g˜)(t, Jt , y), the solution will be a time-inhomogeneous (Ft )strong Markov process. ˆ To prove the above results one needs to start with a solution of (HM P )aJˆ,0b , say, and develop the stochastic integration results in Section V.3 with respect to J. This

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313

general construction is carried out in Section 2 of Perkins (1995) and more general stochastic integrals for “historical semimartingales” with jumps may be found in Evans and Perkins (1998), although a general construction has not been carried out to date. Under the above non-degeneracy condition on a ˆ, this then allows one define a historical Brownian motion, K, from J so that (HSE)Z0 ,K holds, just as one can define the Brownian motion, B, from the solution, X, of dX = σ(X)dB + b(X)dt. Consider next the problem of interactive branching rates. If γ : MF (Rd ) × Rd → R+ , then the extension of (M P )a,b X0 which incorporates this state-dependent branching rate is t 2 d X ∀φ ∈ C (M P )a,b,γ (R ) M (φ) = X (φ) − X (φ) − Xs (AXs φ)ds is a t 0 b t X0 0 t continuous (Ft )-martingale such that M X (φ)t = Xs (γ(Xs )φ2 )ds. 0

In general, uniqueness in law of X remains open. In the context of Fleming-Viot processes Dawson and March (1995) were able to use a dual process for the moments to resolve the analogous problem in which the sampling rates γ(µ, x, y) of types x and y may depend on the population µ in a smooth manner. Their result is a perturbation theorem analogous to that of Stroock and Varadhan (1979) for finitedimensional diffusions but the rigidity of the norms does not allow one to carry out the localization step and so this very nice approach (so far) only establishes uniqueness for sampling rates which are close enough to a constant rate. For our measure-valued branching setting, particular cases of state-dependent branching rates have been treated by special duality arguments (recent examples include Mytnik (1998) and Dawson, Etheridge, Fleischmann, Mytnik, Perkins, Xiong (2000a, 2000b)). If we replace Rd with the finite set {1, . . . d} and the generator Aµ with the state dependent Q-matrix (qij (x)), the solutions to the above martingale problem will be solutions to the stochastic differential equation (V.5.2)

dXtj =

6

2γj (Xt )Xtj dBj (t) +

d 

Xti qij (Xt )dt.

i=1

Some progress on the uniqueness of solutions to this degenerate sde has recently been made by Athreya, Barlow, Bass, and Perkins (2000). ˆ H → (0, ∞) is (Fˆ H )-predictable, then conditions on γˆ are given in If γˆ : R+ × Ω t ˆ

ˆ ,ˆ a,b,ˆ g Perkins (1995) under which (HM P )γτ,J is well-posed. In this martingale problem 0 we have of course t M (φ)t = γˆ (s, J, y)φ(s, y)2Js (dy)ds, φ ∈ Df d . τ

Although the precise condition is complicated (see p. 48-49 in Perkins (1995)), it basically implies that γˆ should be represented by a (possibly stochastic) time integral. It is satisfied in the following examples. Example V.5.3. (a) γˆ (t, J, y) = γ(t, y(t)) for γ ∈ Cb1,2 (R+ × Rd ), γ bounded away from zero. In this case there is no interaction but branching rates may depend on

314

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space-time location something which our strong equation approach does not directly allow.  

t (b) (Adler’s branching goats). γˆ (t, J, y) = exp − 0 pε (ys −yt )Js (dy  )e−α(t−s) ds . The branching rate at yt is reduced if our goat-like particles have grazed near yt in the recent past.
t (c) (General time averaging). γˆ(t, J, y) = t−ε fε (t − s)γ(s+, J, y)ds, where
ε fε : R+ → R+ is C 1 , supp(fε ) ⊂ [0, ε), 0 fε (s)ds = 1, and γ satisfies its analogue of (HLip). Given the difficulties already present in the finite-dimensional case (V.5.2), ˆ ,ˆ a,ˆ b or (HM P )γτ,J would appear to resolving the uniqueness of solutions to (M P )γ,a,b X0 0 be an interesting open problem (see Metivier (1987)). V.6. Appendix: Proofs of Lemma V.4.5 and (V.4.21) We start by proving (IV.4.21), assuming the validity of Lemma V.4.5, and then address the proof of the latter. Assume Zs , B, PB , and K (s) are as in Lemma V.4.5. Let Is (f ), f ∈ Ds (d, d) ≡ (s) Ds refer to the stochastic integral with respect K (s) on (Ω, F , Ft , P). There is some possible confusion here because of the other probabilities PB . Note, however, that if Is,B (f ), f ∈ Ds,B denotes the integral under PB , then Ds ⊂ Ds,B and for f ∈ Ds we may take Is,B (f ) = Is (f ).

(V.6.1)

Let I(f ), f ∈ D continue to denote the stochastic integral with respect to K. The expression “K (s) -a.e.” will always mean with respect to P (not PB ). With these clarifications, and the notation φ(s) introduced prior to Lemma V.4.5, we have: ˆ → Rd is B × Fˆ ∗ -measurable, then Lemma V.6.1. (a) If ψ : R+ × Ω ψ (s) = 0 K − a.e. iff ψ = 0

K (s) − a.e.

(s)

(b) If f ∈ Ds , then f (s) ∈ D and Is (f ) = I(f (s) ) K − a.e. Proof. (a) is a simple exercise in using the definitions. (b) The same is true for the first implication in (b). To check the equality, first let f (t, ω, y) =

n 

fi (ω, y)1(ti−1,ti ] (t) + f0 (ω, y)1{0}(t),

1

where fi ∈ bFˆt∗i +s , 0 = t0 < t1 . . . < tn ≤ ∞. Then (V.6.2)

˜ I (s) s (f, t, ω, y) = Is (f, t − s, ω, Zs (ω, y))1(t ≥ s) n  fi (ω, Z˜s (ω, y)) · [Z˜s (ω, y)((t − s) ∧ ti ) = 1

− Z˜s (ω, y)((t − s) ∧ ti−1 )]1(t ≥ s) =

n  1

fi (ω, Z˜s (ω, y)) · [y(t ∧ (ti + s)) − y(t ∧ (ti−1 + s))].

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315

We also have f (s) (t, ω, y) =

n 

fi (ω, Z˜s (ω, y))1(s+ti−1,s+ti ] (t) + f0 (ω, Z˜s (ω, y))1{s}(t),

1

and so the required result follows for such a simple f from the definition of I(f (s)) and (V.6.2). 0 (s) If f ∈ Ds , then as in (V.3.3) there are (Ft )-simple functions {fk } such that   k  (s) fk (t) − f (t)2 dt > 2−k P Kk < 2−k . 0

(s)

We know f (s) , fk ∈ D by the above and therefore for k ≥ s,   k  (s) fk (t) − f (s) (t)2 dt > 2−k P Kk 0



(s)

= P Kk−s 

(s)

≤ P Kk





k−s

0 k

fk (t) − f (t)2 dt > 2−k

fk (t) − f (t)2 dt > 2−k



 (use Remark V.2.5 (a))

0

< 2−k . A double application of Proposition V.3.2 (d) now allows us to prove the required equality by letting k → ∞ in the result for fk . Proof of (V.4.21). Recall that (X, Z) is the solution of (SE)Z0 ,K . This gives us the Zs which is used to define K (s) and φ(s) . Note that a solution to (SE)Zˆ0 ,K (s) with respect to P will also be a solution with respect to PB (by (V.6.1)) and so we may assume that B = Ω. ˆ Z) ˆ to By Lemma V.4.5 and Theorem V.4.1 (a) there is a unique solution (X, (s) (SE)Zˆ0 ,K (s) on (Ω, F , Ft , P). Define * if t < s Zt (ω, y) Zt (ω, y) = ˆ (s) Zt (ω, y) if t ≥ s, and Xt (ω) = If V (t) =


t 0

*

Xt (ω) if t < s ˆ t−s (ω) if t ≥ s ≡ X

*

if t < s Xt (ω) ˆ (s) (ω) if t ≥ s . X t

ˆ u , Zˆu )du, then by (SE) ˆ (s) , b(X Z0 ,K ˆ Z), ˆ t) + V (t) K (s) − a.e., ˆ = y0 + Is (σ(X, Z(t)

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and so Lemma V.6.1 implies that K-a.e., ˆ Z), ˆ t) + V (s) (t) Zˆ (s) (t) = Zs + Is(s) (σ(X, t−s ˆ Z) ˆ (s) , t) + ˆ u (ω), Zˆu (ω, Z˜s ))du I(t ≥ s) b(X = Zs + I(σ(X, 0 t ˆ (s), Zˆ (s) )1(· ≥ s), t) + ˆ u(s), Zˆu(s) )du 1(t ≥ s). b(X = Zs + I(σ(X s

It follows that K-a.e. for t ≥ s, t b(Xu , Zu )du Z  (t) = Zs + I(σ(X· , Z· ) 1(· ≥ s), t) + s t t   σ(Xu, Zu )dy(u) + b(Xu , Zu )du. = Zs + s

s

Therefore we see that K-a.e., Z  (t) = Z0 +



t

σ(Xu , Zu )du +

0



t

b(Xu , Zu )du,

0

first for t ≥ s by the above, and then for all t ≥ 0 by the fact that (X, Z) solves (SE)Z0 ,K . Also P-a.s. for all t ≥ s (s)  Xt (·) = 1(Zˆt−s ∈ ·)Kt−s (dy) = 1(Zˆt−s (ω, Z˜s (ω, y)) ∈ ·)Kt (dy) = 1(Zt (ω, y) ∈ ·)Kt (dy), and the above equality is trivial for t < s. We have shown that (X  , Z  ) solves ˆ t for all t ≥ 0 a.s. and (SE)Z0 ,K and so X  = X a.s. This means that Xt+s = X (V.4.21) is proved. Proof of Lemma V.4.5. We will show that (V.6.3)

(s)

Kt

(s)

satisfies (HM P )0,K (s) on (Ω, F , Ft , P). 0

It follows immediately that the same is true with respect to PB and so the first result follows. The last inequality then follows trivially. ˆ and Assume first that Zs (ω, y) = Zs (y), Zs ∈ bCs . If φ ∈ D(A)
t ˆ n(t, y) = φ(t, y) − 0 Aφ(r, y)dr, then for t ≥ s, t ˆ (s) (r, y)dr. φ(s) (t, y) = n(t − s, Z˜s (y)) + (Aφ) s

Superprocesses

317

If y0 ∈ C s , s ≤ u ≤ t, and G = {y : y s ∈ F1 , θˆs (y)u−s ∈ F2 } for some Fi ∈ C, then G ∈ Cu and Ps,y0 (n(t − s, Z˜s )1G ) = 1F1 (y0 )Ps,y0 (n(t − s, Zs (y0 ) + θˆs (·))1F2 (θˆs (·)u−s )) = 1F1 (y0 ) n(t − s, Zs (y0 ) + y)1F2 (y u−s )dP 0 (y) = 1F1 (y0 ) n(u − s, Zs (y0 ) + y)1F2 (y u−s )dP 0 (y) = Ps,y0 (n(u − s, Z˜s )1G ), reversing the above steps in the last line. This shows that for any ms ∈ MFs (C), ˆ (s) . Now apply Proposition V.2.6 to the φ(s) ∈ D(A¯s,ms ) and A¯s,ms (φ(s) ) = (Aφ) historical Brownian motion {Kt : t ≥ s} to conclude (s) (s) Kt (φt ) = φs+t Ks+t (dy) s+t s+t (s) ˆ (s) (V.6.4) ) + (r, y)dM (r, y) + Kr ((Aφ) φ = Ks (φ(s) s r )dr. s

s

That is, s+t ˜ ˜ φ(r − s, Z˜s (y))dM (r, y) φt (Zs (y))Kt+s (dy) = φ0 (Zs (y))Ks(dy) + s s+t ˆ (V.6.5) Aφ(r − s, Z˜s (y))Kr (dy)dr ∀t ≥ 0 a.s. + n

s

i i i If Zs (ω, y) = i=1 1Bi (ω)Z (y) for Bi ∈ Fs and Z ∈ bCs , set Zs = Z in (V.6.5), multiply by 1Bi (ω) and sum over i to see that (V.6.5) remains valid if Zs (y) is replaced by Zs (ω, y). Now pass to the pointwise closure of this class of Zs (ω, y) ˆ to conclude that (V.6.5) remains valid if Zs is and use the continuity of φ and Aφ Fˆs∗ -measurable (Remark V.2.5 (c) allows us to pass from Fˆs to Fˆs∗ on the right-hand side of (V.6.5) for all t ≥ 0 simultaneously). Now reinterpret this general form of (V.6.5) as (V.6.4) and let M (s) (φ)(t) denote the stochastic integral in (V.6.4). Then (s) M (s) (φ)(t) is an (Ft )-martingale and s+t t 2 Kr (φ(s) )dr = Kr(s) (φ2r )dr. M (s) (φ)t = r s

0

The last term on the right-hand side of (V.6.5) equals first term equals

(s) K0 (φ0 ).


t 0

(s)

ˆ r ) dr and the Kr ((Aφ)

This proves (V.6.3) and so completes the proof.

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A. Etheridge and P. March (1991). A note on superprocesses. Probab. Th. Rel. Fields 89, 141-147. S.N. Ethier and T.G. Kurtz (1986). Markov Processes: Characterization and Convergence. Wiley, N.Y. S.N. Ethier, and T.G. Kurtz (1993). Fleming-Viot processes in population genetics. Siam. J. Cont. Opt. 31, 345-386. S.N. Evans and E. Perkins (1991). Absolute continuity results for superprocesses. Trans. Amer. Math. Soc. 325, 661-681. S.N. Evans and E. Perkins (1994). Measure-valued branching diffusions with singular interactions. Can. J. Math. 46, 120-168. S.N. Evans and E. Perkins (1998). Collision local times, historical stochastic calculus and competing species. Elect. J. Prob. 3, paper 5. W. Feller (1939). Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in Wahrscheinlichkeitstheoretischer Behandlung. Acta Biotheoretica 5, 1140. W. Feller (1951). Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Prob., Univ. of California Press Berkeley, pp. 227-246. P.J. Fitzsimmons (1988). Construction and regularity of measure-valued Markov processes. Israel J. Math. 64, 337-36. P.J. Fitzsimmons (1992). On the martingale problem for measure-valued branching processes. In: Seminar on Stochastic Processes 1991, pp. 39-51, Birkh¨ auser, Boston. I.I. Gihman and A.V. Skorohod (1975). The Theory of Stochastic Processes II. Springer, New York. T.E. Harris (1963). The Theory of Branching Processes. Springer, Berlin. R. van der Hofstad, G. Slade (2000). Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Preprint. D.N. Hoover and H.J. Keisler (1984). Adapted probability distributions. Trans. Amer. Math. Soc. 286, 159-201. D.N. Hoover and E. Perkins (1983). Nonstandard construction of the stochastic integral and applications to stochastic differential equations I, II, Trans. Amer. Math. Soc. 275, 1-36, 37-58. N. Ikeda and S. Watanabe (1981). Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam. I. Iscoe (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Th. Rel. Fields 71, 85-116. I. Iscoe (1988). On the supports of measure-valued critical branching Brownian motion. Ann. Prob. 16, 200-221. J. Jacod (1977). A general theorem of representation for martingales. In: Proceedings of Symposia in Pure Math. 31, pp. 37-54, Amer. Math. Soc, Providence.

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J. Jacod and A.N. Shiryaev (1987). Limit theorems for Stochastic Processes. SpringerVerlag, Berlin. A. Jakubowski (1986). On the Skorohod topology. Ann. Inst. H. Poincar´e B22, 263-285. S. Kakutani (1944). On Brownian motion in n-space. Proc. Imp. Acad. Tokyo 20, 648-652. O. Kallenberg (1986). Random Measures. Academic Press, New York. F. Knight (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence. N. Konno and T. Shiga (1988). Stochastic differential equations for some measurevalued diffusions. Probab. Th. Rel. Fields 79, 201-225. N. Krylov (1997). On a result of Mueller and Perkins. Probab. Th. Rel. Fields 108, 543-557. N. Krylov (1997b). On SPDE’s and superdiffusions. Ann. Prob. 25, 1789-1809. T.G. Kurtz (1975). Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618-642. T.G. Kurtz (1998). Martingale problems for conditional distributions of Markov processes. Elect. J. Prob. 3, paper 9. J.F. Le Gall (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Th. Rel. Fields 95, 25-46. J.F. Le Gall (1993b). The uniform random tree in a Brownian excursion. Probab. Th. Rel. Fields 96, 369-383. J. F. Le Gall (1994). A lemma on super-Brownian motion with some applications. In: The Dynkin Festschrift pp. 237-251. Birkh¨ auser, Boston. J. F. Le Gall (1998). The Hausdorff measure of the range of super-Brownian motion. In: Perplexing Problems in Probability–Festschrift in Honor of Harry Kesten, pp. 285-314, Birkh¨auser, Boston. J. F. Le Gall (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zurich, Birkh¨ auser Verlag, Basel. J. F. Le Gall and E. Perkins (1995). The Hausdorff measure of the support of two-dimensional super-Brownian motion. Ann. Prob. 232, 1719-1747. J. F. Le Gall, E. Perkins and S.J. Taylor (1995). The packing measure of the support of super-Brownian motion. Stoch. Process. Appl. 59, 1-20. Z. Li and T. Shiga (1995). Measure-valued branching diffusions: immigrations, excursions and limit theorems, J. Math. Kyoto Univ. 35, 233-274. M. Lopez (1996). Path properties and convergence of interacting superprocesses. Ph.D. thesis, UBC.

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M. Metivier (1987). Weak convergence of measure-valued processes using Sobolevimbedding techniques. In: Proc. Trento 1985 SPDE and Applications, Lect. Notes. Math. 1236, 172-183, Springer, Berlin. C. Mueller and E. Perkins (1992). The compact support property for solutions of the heat equation with noise. Probab. Th. Rel. Fields 93, 325-358. C. Mueller and R. Tribe (1994). Stochastic pde’s arising from the long range contact and long range voter processes. Probab. Th. Rel. Fields 102, 519-546. L. Mytnik (1998). Weak uniqueness for the heat equation with noise. Ann. Prob. 26, 968-984. L. Mytnik (1999). Uniqueness for a competing species model. Can. J. Math. 51, 372-448. A. Pazy (1983). Semigroups of Linear Operators and Applications to Partial Differential Applications. Springer-Verlag, Berlin. E. Perkins (1988). A space-time property of a class of measure-valued branching diffusions. Trans. Amer. Math. Soc. 305, 743-795. E. Perkins (1989). The Hausdorff measure of the closed supoprt of super-Brownian motion. Ann. Inst. H. Poincar´e Stat. 25, 205-224. E. Perkins (1990). Polar sets and multiple points for super-Brownian motion. Ann. Prob. 18, 453-491. E. Perkins (1991). On the continuity of measure-valued processes. In: Seminar on Stochastic Processes 1990, pp. 261-268, Birkh¨ auser, Boston. E. Perkins (1992). Measure-valued branching diffusions with spatial interactions. Probab. Th. Rel. Fields 94, 189-245. E. Perkins (1992b). Conditional Dawson-Watanabe processes and Fleming-Viot processes. In: Seminar on Stochastic Processes 1991, pp.142-155, Birkh¨ auser, Boston. E. Perkins (1994). The Strong Markov Property of the Support of Super-Brownain Motion. In: The Dynkin Festschrift Markov Processes and their Applications, pp.307-326, Birkh¨ auser, Boston. E. Perkins (1995). On the martingale problem for interactive measure-valued diffusions. Mem. Amer. Math. Soc. 115 n. 549. E. Perkins (1995b). Measure-valued branching diffusions and interactions. Proceedings of the International Congress of Mathematicians, Zurich, 1994, pp.1036-1046, Birkh¨ auser Verlag, Basel. E. Perkins and S.J. Taylor (1998). The multifractal spectrum of super-Brownian motion. Ann. Inst. H. Poincar´e Stat. 34, 97-138. S.C. Port and C. J. Stone (1978). Brownian Motion and Classical Potential Theory. Academic, New York. M. Reimers (1989). One-dimensional stochastic partial differential equations and the branching measure-diffusion. Probab. Th. Rel. Fields 81, 319-340.

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M. Reimers (1989b). Hyperfinite methods applied to the critical branching diffusion. Probab. Th. Rel. Fields 81, 11-27. D. Revuz and M. Yor (1991). Springer, Berlin.

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S. Roelly-Coppoletta (1986). A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17,43-65. L.C.G. Rogers and D. Williams (1987). Diffusions, Markov Processes and Martingales Vol. 2. Wiley, Chichester. C. A. Rogers and S. J. Taylor (1961). Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1-31. H. Rost (1971). The stopping distributions of a Markov process. Invent. Math. 14, 1-16. W. Rudin (1974). Real and Complex Analysis. McGraw-Hill, New York. A. Schied (1996). Sample path large deviations for super-Brownian motion. Probab. Th. Rel. Fields 104, 319-347. M. Sharpe (1988). General Theory of Markov Processes. Academic, Boston. T. Shiga (1990). A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto. Univ. 30, 245-279. D.W. Stroock and S.R.S. Varadhan (1979). Multidimensional Diffusion Processes, Springer-Verlag, Berlin. S. Sugitani (1987). Some properties for the measure-valued diffusion process. J. Math. Soc. Japan 41, 437-462. A. Szulga (1982). On minimal metrics in the space of random variables. Teor. Veroyatn. Primen 27, 401-405. S.J. Taylor (1961). On the connections between generalized capacities and Hausdorff measures. Proc. Cambridge Philos. Soc. 57, 524-531. S.J. Taylor (1964). The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cam. Phil. Soc. 60, 253-258. S.J. Taylor and N.A. Watson (1985). A Hausdorff measure classification of polar sets for the heat equation. Math. Proc. Cam. Phil. Soc. 47, 325-344. R. Tribe (1991). The connected components of the closed support of super-Brownian motion. Probab. Th. Rel. Fields 84, 75-87. R. Tribe (1992). The behavior of superprocesses near extinction. Ann. Prob. 20, 286-311. R. Tribe (1994). A representation for super Brownian motion, Stoch. Proc. Appl. 51, 207-219. J. Walsh (1986). An Introduction to Stochastic Partial Differetial Equations, Ecole d’Et´e de Probabilit´es de Saint Flour 1984, Lect. Notes. in Math. 1180, Springer, Berlin.

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S. Watanabe (1968). A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto U. 8, 141-167. T. Yamada and S. Watanabe (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto U. 11, 155-167.

Index absolute continuity of superprocesses with distinct drifts or selection schemes,252–255 of superprocesses with distinct initial conditions, 203, 204 balayage,159, 160 Borel strong Markov process (BSMP),137 branching particle system,143-147 C-relatively compact,148 Campbell measure,214, 187 canonical measures,178 convergence theorems for, 180, 182 for superprocesses, 180, 183 for historical processes, 192 capacity,224 charging sets,223 collision local time,263 for a class of interacting measure-valued processes, 269 for super-Brownian motion, 263 Tanaka formula, 270 compact containment for branching particle systems,155 for measure-valued processes, 157 compact support property,198, 199 competing species model,247 in higher dimensions, 278 in one dimension, 257–262 martingale problem, 248, 277 non-existence in dimensions greater than 3, 279 conditioning on the total mass process,172, 207 contact process,134 long range limit in one dimension, 262 degenerate stochastic differential equation,313 domination principle,250, 177 exchangeable particle representation,310–311 excursion measures for superprocesses,182 exit measures for super-Brownian motion,225 exponential moments for super-Brownian motion,209 extinction point,204,206–207 extinction probabilities for branching processes,135 for superprocesses, 171 Feller branching diffusion,137 convergence of branching processes to, 137, 159–160 Feller process,142 Feynmann-Kac formula for Markov processes,167

326

Superprocesses

for path-valued processes, 190 for solutions of nonlinear pde, 170, 138 first moment formula for superprocesses,167 Fleming-Viot process,256 almost as a time-change of a normalized superprocess, 206 as a limit of measure-valued branching with finite carrying capacity,256–257 with interactions, 313 forward equation,311 Galton-Watson branching process,135 generator,137, 138 Brownian, 140 core, 140 space time, 166 symmetric stable, 140 goats,314 graph of a measure-valued process,263 Green function representation for superprocesses,167 Hausdorff measure and capacity,224 definition, 207 of the range of super-Brownian motion, 208, 213, 233 of the support of super-Brownian motion, 207, 209, 212 of the support of the super-symmetric stable processes, 213 restricted, 274 Hausdorff metric,244 historical Brownian motion,194 extended martingale problem, 289 mean measure, 283 martingale problem using functions of finitely many coordinates, 291 stochastic integration along, 292–296 historical cluster representation,193, 240 historical martingale problem (HMP),190 historical modulus of continuity,195 historical process canonical measure,192 definition of, 188 for a branching particle system, 146 Laplace functional equation, 191 Markov property, 190 martingale problem, 190 mean measure, 191 relationship to superprocesses, 189, 191 weak convergence to, 189 hitting probabilities for super-Brownian motion,228 at a fixed time, 237–239 hitting sets,223

Superprocesses

327

Hunt process,137 infinitely divisible random measures,178 canonical representation of, 178, 181 and DW-superprocesses, 178 canonical measures, 178 inhomogeneous Borel strong Markov process (IBSMP),188 instantaneous support propagation, see support propagation interactive measure-valued branching diffusions Markov and Feller properties,301–301 pathwise existence and uniqueness,300 stability results, 301 uniqueness in law for the martingale problem, 310,313 intersections of the supports of two independent super-Brownian motions,239 intersections of the graphs of two super-Brownian motions,263-264 intersections between the graphs of a Brownian and super-Brownian motion,278 Itˆo’s lemma for historical stochastic integrals,298 Kolmogorov’s theorem for branching processes,135, 194 Laplace functional,167 equation for historical processes, 191 equation for superprocesses, 168 local density theorems for super-Brownian motion,179, 210 for super-symmetric stable processes, 214–215 local martingale problem for Dawson-Watanabe superprocesses,159 for time-dependent functions, 164 local time,201 lookdowns,311 marriage lemma,306 martingale problem for a competing species model,277 for interactive branching, 313 for Dawson-Watanabe superprocesses, 149 for historical Brownian motion, 291 for spatial interactions, 310 for spatially interacting historical processes, 313 modulus of continuity for historical Brownian motion,195 for historical processes, 198 for the supports of super-Brownian motion, 200 monotone class lemma,160 multi-indices,144 multiplicative property of superprocesses,168 nonlinear pde,168 asymptotics at infinity, 240 existence, uniqueness and regularity, 169, 172–177 mild form, 169

328

Superprocesses

removable singularities, 225 nonstandard analysis,134 open problems,217, 224, 244, 246, 280, 313 path-valued process,141, 190 domain of generator, 283 extended generator, 290 Poisson point processes and superprocesses,182 polar sets,224 predictable representation property of super-Brownian motion,248 range of super-Brownian motion compactness of,200 definitions, 200 resolvent,138 scaling properties of superprocesses,172 second moment formula for superprocesses,167 snake,132 square function inequality,152 stochastic calculus for historical stochastic integrals,302 stochastic equation for spatial interactions Feller property of solutions,300 for historical processes, 313 for measure-valued branching processes, 282, 300 for particle systems, 282 martingale problem, 310 pathwise existence and uniqueness, 300 stability, 301 strong Markov property of solutions, 301 stochastic integration along historical paths or Brownian trees,192–196 with respect to a martingale measure, 161 with respect to white noise, 215 stochastic pde for the density of one-dimensional super-Brownian motion,126 for a competing species model, 261 for a self-competition model, 262 super-Brownian motion as a function of its support,246 definition, 194 density in one dimension, 216 disconnectedness of the support, 242 discontinuities of the support process, 245 exponential bounds, 209 fixed time hitting estimates, 237–239 Hausdorff measure of the support, 207, 209, 212 Hausdorff measure of the range, 208, 213, 233 hitting probabilities of small balls and points, 228

Superprocesses local density theorem, 209, 210 strong Markov property of support process, 246 packing measure, 212 polar sets, 225, 232 polar sets for the graph, 274 predictable representation property, 248 restricted Hausdorff measure of its graph, 274 singularity of, 212 strong continuity of, 223 super-L´evy processes,203 super-Poisson process,200–201, 206 super-symmetric stable processes,203, 204, 213 supports compactness of,198 disconnectedness of, 242 instantaneous propagation of, 201 instantaneous propagation for diffusions, 200 instantaneous propagation for super-L´evy processes, 203 instantaneous propagation for super-Poisson processes, 206 strong Markov property of, 246 tightness of measure-valued processes,148 of martingales, 152 Tanaka formula for the graphs of a class of measure-valued processes,270 time-change of superprocesses,206 Vasershtein metric,184, 281 voter model,134 white noise,215 Yaglom’s theorem for branching processes,135, 182, 194

329

Part III

Aad van der Vaart:

Semiparametric Statistics

E. Bolthausen, E. Perkins, A. van der Vaart: LNM 1781, P. Bernard (Ed.), pp. 331–457, 2002. c Springer-Verlag Berlin Heidelberg 2002


Table of Contents

1

Lecture: Introduction, Tangent Sets . . . . . . . . . . . . . . . . . . . . . . 336 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 1.2 Tangent Spaces and Information . . . . . . . . . . . . . . . . . . . . . . . . . . 339

2

Lecture: Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 2.1 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 2.2 Efficient Score Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

3

Lecture: Calculus of Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 3.1 Score and Information Operators . . . . . . . . . . . . . . . . . . . . . . . . . 357 3.1.1 Information Loss Models . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3.2 Semiparametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

4

Lecture: Gaussian Approximations . . . . . . . . . . . . . . . . . . . . . . . . 370 4.1 Contiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 4.2 Gaussian Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

5

Lecture: Empirical Processes and Consistency of Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 5.1 Empirical Measures and Entropy Numbers . . . . . . . . . . . . . . . . . 383 5.2 Glivenko–Cantelli Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 5.3 Consistency of M- and Z-estimators . . . . . . . . . . . . . . . . . . . . . . . 386 5.3.1 Trick 1: Errors-in-variables . . . . . . . . . . . . . . . . . . . . . . . . 388 5.3.2 Trick 2: Cox model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.3.3 Trick 3: Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 5.4 Nuisance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

6

Lecture: Empirical Processes and Normality of Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6.1 Weak Convergence in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 395 6.2 Donsker Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 6.3 Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 6.4 Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 6.5 Asymptotic Normality of Z-Estimators . . . . . . . . . . . . . . . . . . . . 406 6.6 Nuisance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

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TABLE OF CONTENTS

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Lecture: Efficient Score and One-step Estimators . . . . . . . . . 412 7.1 Efficient Score Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 7.2 One-step Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 7.3 Symmetric location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 7.4 Errors-in-Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

8

Lecture: Rates of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 8.1 A General Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 8.2 Nuisance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 8.3 Cox Regression with Current Status Data . . . . . . . . . . . . . . . . . 428

9

Lecture: Maximum and Profile Likelihood . . . . . . . . . . . . . . . . 433 9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 9.2 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 9.3 Cox Regression with Current Status Data . . . . . . . . . . . . . . . . . 439 9.4 Profile Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

10 Lecture: Infinite-dimensional Z-Estimators . . . . . . . . . . . . . . . . 446 10.1 General Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 10.2 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Preface These notes provide extended versions of my lectures in the St Flour meeting of 1999. The general subject are semiparametric models for replicated experiments, in particular the theory for functionals that are estimable at the rate equal to the square root of the number of replications. We discuss bounds on the efficiency of estimators and tests, and methods of constructing efficient or inefficient estimators and tests, with particular attention for maximum likelihood estimators. Furthermore, we discuss abstract empirical processes, which play an important role in the analysis of the estimators. The ten lectures have a certain overlap with material earlier published in the books [41] and [42]. A number of proofs have been omitted, because they can be found in these works. On the other hand, these notes are an attempt to give a consistent and reasonably self-contained overview of (a part of) semiparametric statistics, including digressions into empirical process theory, new examples, and a number of more recent developments. This area is certainly not complete. To illustrate this point, scattered through the text we pose some problems whose solutions are presently unknown (to me). Our list of references is restricted to the references that are directly relevant to the lectures. In beginning 2000 the Mathematical Reviews gave 415 responses to a query on semiparametric models, so our list does not do justice to the great amount of work having been done. A general work covering the subject of semiparametric models, but from a somewhat different point of view with relatively little attention for the subject of Lectures 5–10, is the book [3] by Bickel, Klaassen, Ritov and Wellner. This book also has an extensive list of references.

Notation We use the wiggly arrow
for weak convergence, also for nonmeasurable maps: if Xn and X are maps defined on some probability spaces (Ωn , Un , Pn ) with values in a metric space D, then we say that Xn
X if E∗ f (Xn ) → Ef (X) for all bounded, continuous functions f : D → R. Here the limit X is always assumed Borel measurable, but the Xn may be arbitrary maps. The ∗ in E∗ f (Xn ) is for outer expectation on (Ωn , Un , Pn ). Given a measure space (X , A, P ) the set Lr (P ) (for
r ≥ 1) is the collection of all measurable functions f : X → R with f rP,r : = |f |r dP < ∞. The wiggly inequality  means “less than equal up to a constant”. The range and kernel of an operator A are denoted by R(A) and N(A). The space of all bounded functions z: T → R on a set T is denoted by ∞ (T ) and zT is the uniform norm. The set U C(T, ρ) is the set of all ρ-uniformly continuous functions on T .

1. Lecture: Introduction, Tangent Sets

In this lecture we introduce basic notation, give a number of examples of semiparametric models, and define the tangent set of a model.

1.1 Introduction Throughout the presentation of the general theory we denote by X1 , . . . , Xn the observations. These are measurable maps on some underlying probability space that we usually need not further specify, and take values in a measurable space (X , A). The observations are independent and identically distributed (i.i.d.), with a distribution P on (X , A). A model P is a collection of probability measures on the sample space, to be considered the set of all possible values of P . A semiparametric model is one that is neither a parametric model nor a nonparametric model. This definition is not informative, but could be saved by giving precise definitions of parametric and nonparametric models. The nonparametric model P is the set of all probability distributions on P. A parametric model is a model that can be smoothly indexed by a Euclidean vector (“the parameter”). We shall not attempt to make this definition more precise by specifying “smoothly”, but note that this should cover all classical statistical models, including exponential families and the uniform distributions. The concept of a “nonparametric model” is often also used in a more vague sense of a model that does not essentially restrict the elements P ∈ P. A model in which all P are assumed to have a second moment or a smooth density relative to Lebesgue measure is then also considered to be “nonparametric”. Thus the “definition” says that a semiparametric model is an infinitedimensional model that is essentially smaller than the set of all possible distributions. Even this vague description is not universally accepted. For instance: the nonparametric model is often considered to be semiparametric if it is parametrized in an interesting way. A few examples will give a better idea. Example 1.1 (Symmetric location). For a given θ ∈ R and a probability density η on R that is symmetric about 0, let Pθ,η be the measure with density x → η(x − θ). Then consider the semiparametric model P consisting of all

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337

measures Pθ,η when θ ranges over R and η ranges over all Lebesgue densities that are
absolutely continuous with finite Fisher information for location: I(η): = (η  /η)2 η dλ < ∞. This model arose naturally in the study of nonparametric testing theory (e.g. rank tests) and was studied long before the general subject of semiparametric models had been conceived. It turns out to be a very special model as regards the estimation of the center of symmetry θ. As we shall see there exist estimators for θ in this model (which cannot use the form of the unknown η) that are (asymptotically) of the same quality as the best estimators specially designed to work for a particular η (for instance as good as the sample mean in the case of normal η and as good as the median for Laplace η). Example 1.2 (Partially linear regression). A classical regression model specifies that the conditional mean of a “response variable” Y given a covariate V is a linear function θT V of the covariate, or a fixed transformation Ψ (θT V ) of it. A nonparametric regression model would replace the linear function by an arbitrary function, perhaps restricted by being “smooth”. A typical semiparametric model would mix these two extremes, for instance by specifying that the conditional mean is of the form Ψ (θT V + η(W )) for θ ∈ Rd and η ranging over the class of all twice differentiable functions on the domain of W , and a fixed function Ψ . To describe the full model we could specify that the observation is X = (Y, V, W ) and that (V, W ) has an arbitrary distribution. Next there are several possibilities to complete the description by specifying the form of the conditional distribution of Y given (V, W ). One possibility is to specify only that E(Y | V, W ) = θT V + η(W ). This type of model is popular among econometricians. A smaller model is obtained by postulating that Y = θT V + η(W ) + e for e independent of (V, W ) and of mean zero, leaving the rest of the distribution of e unspecified, assuming it to be normal or symmetric. Third, we can also create semiparametric versions of the generalized linear model. For instance, the response Y could be a 0-1 variable and we could assume that P(Y = 1| V, W ) is of the form Ψ (θT V + η(W )) for Ψ the logistic distribution function. Example 1.3 (Cox). In the Cox model a typical observation is a pair X = (T, Z) of a “survival time” T and a covariate Z. It is best described in terms of the conditional hazard function of T given Z. Recall that the hazard function λ corresponding to a probability density f is the function λ = f /(1 − F ), for F the distribution function corresponding to f . Simple algebra shows that 1 − F = e−Λ and hence f = λe−Λ , so that the relationship between f and λ is on-to-one. In the Cox model the distribution of Z is arbitrary and the conditional T hazard function of T given Z is postulated to be of the form eθ Z λ(t) for θ ∈ Rd and λ being a completely unknown hazard function. The parameter θ has an interesting interpretation in terms of a ratio of hazards. For

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instance, if the ith coordinate Zi of the covariate is a 0-1 variable then eθi is the ratio of the hazards of two individuals whose covariates are Zi = 1 and Zi = 0, respectively, and whose covariates are identical otherwise. This is one reason for the popularity of the model: the model gives a better fit to data than a parametric model (obtained for instance by assuming that the baseline hazard function is of Weibull form), but its parameters are still easy to interpret. A second reason for its popularity is that statistical procedures for estimating the parameters take a simple form. They were originally found and motivated by ad-hoc arguments. We shall use the model throughout these lectures as an illustration and show how the standard estimators can be derived and analysed by principles that apply equally well to other semiparametric models. Example 1.4 (Mixture models). Suppose that x → pθ (x| z) is a probability density for every pair (θ, z) ∈ Θ × Z for a subset Θ of a Euclidean space and a measurable space (Z, C). If the map (x, z) → pθ (x| z) is jointly measurable, then  pθ,η (x) = pθ (x| z) dη(z) defines a probability density for every probability measure η on (Z, C). This mixture density reduces to the density pθ (·| z) when η is degenerate at z. Hence the model consisting of all mixture densities of this type is considerably bigger than the “original model”, which is parametric if z is Euclidean and the map (θ, z) → pθ (·| z) is smooth. A concrete example of a mixture model is the errors-in-variables model, which is most easily described structurally, as follows. The observation is a pair X = (X1 , X2 ), where X1 = Z + e and X2 = gθ (Z) + f for a bivariate normal vector (e, f ) with mean zero and unknown covariance matrix, and a function gθ that is known up to a parameter θ. Thus X2 is a (possibly nonlinear) regression on a variable Z that is observed with error. The distribution of Z is unknown. The kernel pθ (·| z) is in this case a multivariate Gaussian density. A particular example is the linear errors-in-variables model, for which θ = (α, β) and gθ (z) = α + βz. This linear model has been studied before the 1980s, but not from a semiparametric perspective. Semiparametric theory has led to new, more efficient estimators of the regression parameters. Surprisingly, for most of the nonlinear cases good estimators for θ are still √ unknown, and in fact it is unknown if the parameter θ is estimable at n rate in general. (See [35] and work in progress by the same author.) Example 1.5 (Random censoring). A “time of death” T is observed only if death occurs before the time C of a “censoring event” that is independent of T ; otherwise C is observed. Thus, a typical observation X is a pair of a survival time and a 0-1 variable, and is distributed as (T ∧ C, 1{T ≤ C}). If the distributions of T and C are allowed to range over all distributions

1.2 Tangent Spaces and Information

339

on [0, ∞], then the distribution of X can be shown to take an arbitrary form on the sample space X = [0, ∞) × {0, 1}. Therefore, this model is a nonparametric example. Because the interest is usually in the distribution of T , which is a complicated function of the distribution of X to which much of the semiparametric machinery applies, the model is usually also considered semiparametric. Our lectures aim at developing theory for the estimation and testing of functionals ψ: P → B defined on a model P and taking values in some Banach space B (most often Rd ). An important example is the functional ψ(Pθ,η ) = θ if the model P = {Pθ,η : θ ∈ Θ, η ∈ H} is indexed by two “parameters” θ and η. In this case, because apparently the prime interest is in θ, we refer to η as a “nuisance parameter”. This will not stop us from also considering the estimation of η. Models with a partitioned parameter (θ, η), with θ finitedimensional, are semiparametric models in a strict sense. Begun, Hall, Huang and Wellner in [1] called them parametric-nonparametric, having in mind that η would be an element of a nonparametric model. Our main interest in these lectures is in functionals ψ that allow an asymptotic theory analogous to the theory for smooth parametric models. This comprises the asymptotic normality of the maximum likelihood estimator, rooted in the work by Fisher in the 1920s, the asymptotic chisquare distribution of the likelihood ratio statistic, rooted in the work by Wilks in 1930s, and the lower bound theory rooted in the work by Cram´er and Rao in the 1940s. The dates might suggest that we are only setting out to obtain a simple extension of “classical theory” of the first half of the 20th century. There is some truth to this, but as we shall see, apart from necessitating more mathematical sophistication (which word we mean to use in a positive sense), the theory of semiparametric models turns out to be much richer than the classical theory. Unfortunately, not all problems have been solved. This is true for the problems in the restricted realm of the preceding paragraph. It is even more true for the general theory of semiparametric models, which also contains many so-called inverse problems. In later lectures we shall indicate some of the important open questions, to be solved in the next millennium. In the following section we start by developing a notion of “information” for estimating ψ(P ) given the model P, which extends the notion of Fisher information for parametric models.

1.2 Tangent Spaces and Information To estimate the parameter ψ(P ) given the model P is certainly harder than to estimate this parameter given that P belongs to a submodel P0 ⊂ P. For every smooth parametric submodel P0 = {Pθ : θ ∈ Θ} ⊂ P, we can calculate the Fisher information for estimating ψ(Pθ ). Then the “information” for estimating ψ(P ) in the whole model is certainly not bigger than the infimum of

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the informations over all submodels. We shall simply define the information for the whole model as this infimum. A submodel for which the infimum is taken (if there is one) is called least favourable or a “hardest” submodel. In most situations it suffices to consider one-dimensional submodels P0 . These should pass through the “true” distribution P of the observations, and be differentiable at P in an appropriate way. Definition 1.6. A differentiable path is a map t → Pt from a neighbourhood of 0 ∈ [0, ∞) to P such that, for some measurable function g: X → R,   1/2 2 dPt − dP 1/2 − 12 g dP 1/2 → 0. t The function g is called the score function of the submodel {Pt : t ≥ 0} at t = 0. 1/2

The notation in the preceding display is due to Le Cam. The objects dPt can be formalized by introducing an Hilbert space of “square roots of measures”. Simpler and sufficient for our purposes is to read the display as   1/2 1/2 2 ptt − pt 1/2 − 12 g pt dµt → 0, t where, for each t, µt is an arbitrary measure relative to which P and Pt possess densities pt and ptt . For instance, the measure µt = Pt + P , or a fixed σ-finite dominating measure for P if it exists. The value of the integral does not depend on the choice of µt . In words we say that a differentiable path is a parametric submodel {Pt : 0 ≤ t < ε} that is differentiable in quadratic mean at t = 0 with score function g. Letting t → Pt range over a collection of submodels, we obtain a collection of score functions, which we call a tangent set of the model P at P , and denote by P˙ P . Lemma 1.7. Every score function satisfies P g = 0 and P g 2 < ∞. Proof. For given, arbitrary tn ↓ 0, let pn and p be densities of Ptn and P relative to a σ-finite dominating measure µ, for instance a convex combination √ of the countably many measures Ptn + P . By (1.6) the sequence ( pn − √ 1 √ p)/tn converges in quadratic mean (i.e. in L2 (µ)) to 2 g p. This implies √ immediately that g ∈ L2 (P ). Furthermore, it implies that the sequence pn √ converges in quadratic mean to p. By the continuity of the inner product, √   √  pn − p  √ √ √  1 √ pn + p dµn . Pg = 2 g p 2 p dµ = lim tn The right side equals (1 − 1)/tn = 0 for every n, because both probability densities integrate to 1. Thus P g = 0.

1.2 Tangent Spaces and Information

341

It follows that a tangent set can be identified with a subset of L2 (P ), up to equivalence classes. The tangent set is often a linear space, in which case we speak of a tangent space. Geometrically, we may visualize the model P, or rather the corresponding set of “square roots of measures” dP 1/2 , as a subset of the unit ball of a Hilbert space (the space L2 (µ) if the model is dominated), and P˙ P , or rather the set of all objects 12 g dP 1/2 , as its tangent set. Note however that we have not defined a tangent set to be equal to the set of all score functions g that correspond to some differentiable submodel. For many purposes this “maximal tangent set” is too big, so that we have given ourselves the flexibility of calling any set of score functions a tangent set. The drawback will be that in any result obtained later on we must specify which tangent set we are working with. Usually, we construct the submodels t → Pt such that, for every x, g(x) =

∂ log dPt (x). ∂t |t=0

This pointwise differentiability is not required by (1.6). Conversely, given this pointwise differentiability we still need to be able to apply a convergence theorem for integrals to obtain (1.6). The following lemma solves most examples. Lemma  1.8. If pt is a probability density relative to a fixed measure µ and t
→ pt (x) is continuously differentiable in a neighbourhood of 0 and t → p˙2t /pt dµ is finite and continuous in this neighbourhood, then t → Pt is a differentiable path. The differentiability (1.6) is the correct definition for defining information, because it ensures a type of local asymptotic normality, as shown by the following lemma. Lemma 1.9. If the path t → Pt in P satisfies (1.6), then log

n  dP1/√n i=1

dP

1 (Xi ) = √ g(Xi ) − 12 P g 2 + oP (1). n i=1 n

√ Proof. We adopt the notation  of the preceding  proof, but with tn = 1/ n. The random variable Wni = 2 pn /p(Xi ) − 1 is with P -probability 1 welldefined. By (1.6) n n

√ 2 1 var Wni − √ g(Xi ) ≤ E nWni − g(Xi ) → 0, n i=1 i=1  n  √ √



√ √ 2 pn p dµ − 1 = −n pn − p dµ → − 14 P g 2 . E Wni = 2n i=1

(1.1)

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Therefore, combining the preceding pair of displayed equations, we find n

1 Wni = √ g(Xi ) − 14 P g 2 + oP (1). n i=1 i=1 n

(1.2)

n Next, we express the log likelihood ratio in i=1 Wni through a Taylor expansion of the logarithm. If we write log(1 + x) = x − 12 x2 + x2 R(2x), then R(x) → 0 as x → 0, and log

n  pn i=1

p

n   (Xi ) = 2 log 1 + 12 Wni i=1

=

n

Wni −

i=1

1 4

n

2 Wni

+

i=1

1 2

n

(1.3) 2 Wni R(Wni ).

i=1

2 = As a consequence of the right side of (1.1) , it is possible to write nWni 2 g (Xi ) + Ani for random variables Ani such that E|Ani | → 0. The averages An converge in mean and hence in probability to zero. Combination with the law of large numbers yields n

P 2 P g2 . Wni = (g 2 )n + An →

i=1

By the triangle inequality followed by Markov’s inequality, √       nP |Wni | > ε 2 ≤ nP g 2 (Xi ) > nε2 + nP |Ani | > nε2 ≤ ε−2 P g 2 {g 2 > nε2 } + ε−2 E|Ani | → 0. √   The left side is an upper bound for P max1≤i≤n |Wni | > ε 2 . Thus the By the property sequence max1≤i≤n |Wni | converges to zero in probability.  of the function R, the sequence max1≤i≤n R(Wni ) converges in probability to zero term on the right in (1.3) is bounded by nThe last  as well. 2 max1≤i≤n R(Wni ) i=1 Wni . Thus it is oP (1)OP (1), and converges in probability to zero. Combine to obtain that log

n  pn i=1

p

(Xi ) =

n

Wni − 14 P g 2 + oP (1).

i=1

Together with (1.2) this yields the theorem.

For defining the “information” for estimating ψ(P ), only those submodels t → Pt along which the parameter t → ψ(Pt ) is differentiable are of interest. A minimal requirement is that the map t → ψ(Pt ) be differentiable at t = 0, but we need more.

1.2 Tangent Spaces and Information

343

Definition 1.10. A map ψ: P → B is differentiable at P relative to a given tangent set P˙ P if there exists a continuous linear map ψ˙ P : L2 (P ) → B such that for every g ∈ P˙ P and a submodel t → Pt with score function g, ψ(Pt ) − ψ(P ) → ψ˙ P g. t This definition requires that the derivative of the map t → ψ(Pt ) exists in the ordinary sense, and also that it has a special representation. (The map ψ˙ P is much like a Hadamard derivative of ψ viewed as a map on the space of “square roots of measures”.) Our definition is also relative to the submodels t → Pt , but we speak of “relative to P˙ P ” for simplicity. In the case that B = Rk the Riesz representation theorem for Hilbert spaces allows us to write the derivative map ψ˙ P in the form of an inner product. Precisely, there exists a fixed vector-valued, measurable function ψ˜P : X → Rk ,  ψ˙ P g = ψ˜P , gP = ψ˜P g dP. The function ψ˜P is not uniquely defined by the functional ψ and the model P, since only inner products of ψ˜P with elements of the tangent set are specified, and the tangent set does not span all of L2 (P ). However, it is always possible to find a candidate ψ˜P whose coordinate functions are contained in lin P˙ P , the closure of the linear span of the tangent set. This function is unique, and is called the efficient influence function. It can be found as the projection of any other “influence function” onto the closed linear span of the tangent set. Here an influence function will be any measurable function ψ˙ P : X → R whose projection on lin P˙ P is the efficient influence function. In the preceding set-up the tangent sets P˙ P are made to depend both on the model P and the functional ψ. We do not always want to use the “maximal tangent set”, which is the set of all score functions of differentiable submodels t → Pt , because the parameter ψ may not be differentiable relative to it. According to our definition every subset of a tangent set a tangent set itself. The maximal tangent set is a cone: if g ∈ P˙ P and a ≥ 0, then ag ∈ P˙ P , because the path t → Pat has score function ag when t → Pt has score function g. It is rarely loss of generality to assume that the tangent set we work with is a cone as well. Example 1.11 (Parametric model). Consider a parametric model with parameter θ ranging over an open subset Θ of Rm given by densities pθ with respect to some measure µ. Suppose that there exists a vector-valued measurable map ˙θ such that, as h → 0, 

1/2   1/2 1/2 2 pθ+h − pθ − 12 hT ˙θ pθ dµ = o h2 .

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  Then a tangent set at Pθ is given by the linear space hT ˙θ : h ∈ Rm spanned by the score functions for the coordinates of the parameter θ. If the Fisher information matrix Iθ = Pθ ˙θ ˙Tθ is invertible, then every map χ: Θ → Rk that is differentiable in the ordinary sense as a map between Euclidean spaces is differentiable as a map ψ(Pθ ) = χ(θ) on the model relative to the given tangent space. This follows because the submodel t → Pθ+th has score hT ˙θ and

   ∂ χ(θ + th) = χ˙ θ h = Pθ χ˙ θ Iθ−1 ˙θ hT ˙θ . ∂t |t=0 This equation shows that the function ψ˜Pθ = χ˙ θ Iθ−1 ˙θ is the efficient influence function. Example 1.12 (Nonparametric model). Suppose that P consists of all probability laws on the sample space.
Then a tangent
set at P consists of all measurable functions g satisfying g dP = 0 and g 2 dP < ∞. Since a score function necessarily has mean zero, this is the maximal tangent set. It suffices to exhibit suitable one-dimensional submodels. For a bounded function g, consider for instance the exponential family pt (x) = c(t) exp(tg(x)) p0 (x)   or, alternatively, the model pt (x) = 1 + tg(x) p0 (x). Both models have the property that, for every x, g(x) =

∂ log pt (x). ∂t |t=0

By a direct calculation or by using Lemma 1.8, we see that both models also have score function g at t = 0 in the L2 -sense (1.6). For an unbounded function g, these submodels are not necessarily well-defined. However, the  models have the common structure pt (x) = c(t) k tg(x) p0 (x) for a nonnegative function k with k(0) = k  (0) = 1. The function k(x) = 2(1 + e−2x )−1 is bounded and can be used with any g. Example 1.13 (Cox model). The density of an observation in the Cox model takes the form θT z T (t, z) → e−e Λ(t) λ(t) eθ z pZ (z). Differentiating the logarithm of this expression with respect to θ gives the score function for θ, with x = (t, z), T ˙θ,Λ (x) = z − zeθ z Λ(t).

We can also insert appropriate parametric models s → λs and differentiate with respect to s. If a is the derivative of log λs at s = 0, then the corresponding score for the model for the observation is

1.2 Tangent Spaces and Information

Bθ,Λ a(x) = a(t) − eθ

T

345

 z

a dΛ. [0,t]

Finally, scores for the density pZ are functions b(z). The tangent space contains the linear span of all these functions. Note that the scores for Λ can be found as an “operator” working on functions a. Notes Tangent spaces of statistical models as presented here were popularized as a general theory by Pfanzagl in [28], except that Pfanzagl initially did not define differentiable paths through root-densities, which is an idea going back to Le Cam in the 1960s (see [15], [16], [18]). The study of tangent spaces and information in infinite-dimensional models goes further back to Levit and Koshevnik and Levit (see [20] and [19]) in the mid 1970s, who however considered mostly nonparametric models. We are going to use the Cox model as an illustration throughout the ten lectures. Cox introduced it in [7] and discussed the partial likelihood methods of estimation in [8].

2. Lecture: Lower Bounds

In this lecture we state a number of theorems giving lower bounds on the asymptotic performance of estimators and tests, and make these concrete for the estimation of a parameter θ in a strict semiparametric model. Some of the proofs are deferred to Lecture 4.

2.1 Lower Bounds A “lower bound theorem” in statistics is an assertion that something, estimation or testing, cannot be done better than in some way. The best known bound is the Cram´er-Rao bound for the case of independent sampling from a parametric model {Pθ : θ ∈ Θ ⊂ R}, which is taught in most introductory statistics courses. Fact 2.1. If θ → Pθ is differentiable at θ with score function ˙θ and Tn = a differentiable function Tn (X1 , . . . , Xn ) is un unbiased estimator of χ(θ) for √ χ: R → R, then under regularity conditions varθ ( nTn ) ≥ χ (θ)2 /Iθ for Iθ = varθ ˙θ (X1 ) the “Fisher information” for θ. The Cram´er-Rao bound is the number χ (θ)2 /Iθ , which depends solely on the functional χ to be estimated and on the model {Pθ : θ ∈ R}, through its Fisher information. It turns out that this bound is often not sharp, in the sense that there may not exist unbiased estimators Tn for which n−1 their variance is equal to the bound. However, the bound is sharp in a certain asymptotic sense, as n → ∞. One purpose of this lecture is to state the deep theorems that allow a precise formulation of what it means to be “asymptotically sharp”, in a semiparametric context. To motivate the definition of “information” in our semiparametric setup, assume for simplicity that the parameter ψ(P ) is one-dimensional. The Fisher information about t in a differentiable submodel t → Pt with score function g at t = 0 is P g 2 . Thus, the Cram´er-Rao bound for estimating the function t → ψ(Pt ), evaluated at t = 0, is 

dψ(Pt )/dt P g2

2 =

ψ˜P , g2P . g, gP

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The supremum of this expression over all submodels, equivalently over all elements of the tangent set, is a lower bound for estimating ψ(P ) given the model P, when the “true measure” is P . This supremum can be expressed in the norm of the efficient influence function ψ˜P . Lemma 2.2. Suppose that the functional ψ: P → R is differentiable at P relative to the tangent set P˙ P . Then sup g∈lin P˙ P

ψ˜P , g2P = P ψ˜P2 . g, gP

Proof. This is a consequence of the Cauchy–Schwarz inequality (P ψ˜P g)2 ≤ P ψ˜P2 P g 2 and the fact that, by definition, the efficient influence function ψ˜P is contained in the closure of lin P˙ P . We obtain equality by choosing g equal

to ψ˜P . Thus, the squared norm P ψ˜P2 of the efficient influence function plays the role of a “smallest variance”. Similar considerations (take linear combinations) show that the “smallest covariance” for estimating a higherdimensional parameter ψ: P → Rk is given by the covariance matrix P ψ˜P ψ˜PT of the efficient influence function. The following example shows that the Cram´er-Rao parametric set-up is a special case. Example 2.3 (Parametric model). Consider a parametric model as in Example 1.11. If the Fisher information matrix is invertible and the map χ is differentiable, then the efficient influence function is given by ψ˜Pθ = χθ Iθ−1 ˙θ . Thus the appropriate covariance matrix is Pθ ψ˜Pθ ψ˜PTθ = χθ Iθ−1 (χθ )T . This is precisely the Cram´er-Rao bound. It is time to give a precise meaning to “smallest covariance”. We shall state two theorems regarding the estimation problem and one theorem regarding testing. For every g in a given tangent set P˙ P , write Pt,g for a submodel with score function g along which the functional ψ is differentiable. As usual, an estimator Tn is a measurable function Tn (X1 , . . . , Xn ) of the observations. k Definition 2.4.   A function : R → [0, ∞) is subconvex if for all c > 0 the set y: (y) ≤ c is convex, symmetric and closed.

Theorem 2.5 (LAM). Let the functional ψ: P → Rk be differentiable at P relative to the tangent set P˙ P with efficient influence function ψ˜P . If P˙ P is a convex cone, then, for any estimator sequence {Tn } and subconvex function : Rk → [0, ∞),

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√  sup lim inf sup EP1/√n,g  n(Tn − ψ(P1/√n,g) )) ≥ I

n→∞ g∈I



   dN 0, P ψ˜P ψ˜PT .

Here the first supremum is taken over all finite subsets I of the tangent set. The purpose of the theorem is to give a lower bound, depending only on the √ model and the  functional to be estimated, for the liminf of the risk EP  n(Tn − ψ(P ) , for an arbitrary estimator Tn . A ”best” estimator Tn can then be defined as one that attains equality (of the limsup and for every P ∈ P). The theorem is more complicated than that and involves a supremum over the risk over shrinking neighbourhoods of P . A slightly weaker assertion makes this clearer. Let  ·  be the variation norm. Corollary 2.6. inf lim inf

sup

δ>0 n→∞ Q−P <δ

√  EQ  n(Tn − ψ(Q)) ≥



   dN 0, P ψ˜P ψ˜PT .

Without taking the (local) maximum risk the theorem would fail. It is attractive that the LAM theorem applies to any estimator, even though it may blur the distinction between two estimator sequences by evaluating only a maximum risk. The next theorem avoids the maximum, but at the strong price of restricting itself to regular estimator sequences. An estimator sequence Tn is regular at P for estimating ψ(P ) (relative to P˙ P ) if there exists a probability measure L such that  √  n Tn − ψ(P1/√n,g )

P1/√n,g




L,

every g ∈ P˙ P .

It follows from the definition of weak convergence (or the portmanteau lemma), that for a regular estimator sequence and bounded, continuous function  the
limiting local maximum risk in the left side of the
LAMtheorem re- duces to  dL. Thus if  is subconvex, this is bounded by  dN 0, P ψ˜P ψ˜PT for any such . The convolution theorem shows that this discrepancy between limit and lower bound always results from L being more dispersed than the normal measure. Theorem 2.7 (Convolution). Let the functional ψ: P → Rk be differentiable at P relative to the tangent set P˙ P with efficient influence function ψ˜P . Let Tn be regular at P with limit distribution L.
(i) if P˙ P is a cone, then yy T dL(y) − P ψ˜P ψ˜PT is nonnegative definite. cone, then there exists a probability measure M such (ii) if P˙ P is a convex  that L = N 0, P ψ˜P ψ˜PT ∗ M . Both theorems give the message that the normal distribution with mean zero and covariance matrix P ψ˜P ψ˜PT is a best limiting distribution for an estimator sequence. We should not take this in a too absolute sense. For instance,

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shrinkage estimators as first invented by Stein in the 1950s are not regular, and hence are not in the realm of the convolution theorem, and are LAM for certain loss functions , because in fact better than the usual estimators (which are best regular and LAM), but are not asymptotically normal. A second reason to be careful is that both the LAM and convolution theorems need assumptions on the form of the tangent set. Nevertheless, so far best regular estimator sequences have been considered “best” in semiparametric theory. We adopt the same convention in the following definition. Definition 2.8. An estimator sequence is asymptotically efficient at P for estimating the differentiable parameter ψ(P ), if it is regular at P with limit distribution L = N (0, P ψ˜P ψ˜PT ). We note that our definition of asymptotic efficiency is not absolute, because it is relative to a given tangent set, and we permit a variety of tangent sets. In “practice” one hunts for a pair of a tangent set and estimator sequence such that the tangent set is “big enough” and the estimator sequence “efficient enough” so that the latter is asymptotically efficient according to the preceding definition. Next one strongly believes that this is all that need to be said about the problem. The following lemma shows that efficient estimator sequences must be asymptotically approximable by an average of the efficient influence function evaluated at the observations. Because given a sequence y1 , y2 , . . . the difference of the averages y n+1 − y n is proportional to the additional term yn+1 , the lemma explains the name “influence function” for ψ˜P . Lemma 2.9. Let the functional ψ: P → Rk be differentiable at P relative of to the tangent cone P˙ P with efficient influence function ψ˜P . A sequence  estimators Tn is regular at P with limiting distribution N 0, P ψ˜P ψ˜PT if and only if it satisfies n  √  1 ˜ n Tn − ψ(P ) = √ ψP (Xi ) + oP (1). n i=1

(2.1)

Example 2.10 (Empirical distribution). The empirical distribution is an asymptotically efficient estimator if the underlying distribution P of the sample is completely unknown. To give a rigorous expression to this intuitively obvious statement, fix a measurable function f : X → R with P f 2 < n∞, for instance an indicator function f = 1A , and consider Pn f = n−1 i=1 f (Xi ) as an estimator for the function ψ(P ) = P f . In Example 1.12 it is seen that the maximal tangent space for the nonparametric model is equal to the set of all g ∈ L2 (P ) such that P g = 0. For a general function f , the parameter ψ may not be differentiable relative to the maximal tangent set, but it certainly is differentiable relative to the tangent space consisting of all bounded, measurable functions g with P g = 0. The closure of this tangent space is the maximal tangent set and

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hence working with this smaller set does not change the efficient influence functions. For a bounded function g we can use the submodel defined by dPt = (1 + tg) dP , for which ψ(Pt ) = P f + tP f g. Hence the derivative of ψ is the map g → ψ˙ P g = P f g and the efficient influence function relative to the maximum tangent set is the function ψ˜P = f − P f . (The function f is an influence function; its projection onto the closed, linear span of P˙ P is f − P f .) The “optimal asymptotic variance” for estimating P → P f is equal to P ψ˜P2 = P (f − P f )2 . The sequence of empirical estimators Pn f is asymptotically efficient, because it satisfies (2.1), with the oP (1)-remainder term identically zero. The problem of testing a null hypothesis H0 : ψ(P ) ≤ 0 versus the alternative H1 : ψ(P ) > 0 is closely connected to the problem of estimating the function ψ(P ). It ought to be true that a test based on an asymptotically efficient estimator of ψ(P ) is, in an appropriate sense, asymptotically optimal. For real-valued parameters ψ(P ) this optimality can be taken in the absolute sense of an asymptotically (locally) uniformly most powerful test. For higher-dimensional parameters it is difficult to define a satisfactory notion of asymptotic optimality. We therefore first concentrate on real-valued functionals ψ: P → R. Given a model P and a measure P on the boundary of the hypotheses, i.e. ψ(P ) = 0, we shall study the “local asymptotic power” in a neighbourhood of P . For every score function g for which ψ˙ P g = P ψ˜P g > 0, the corresponding submodel Pt,g belongs to the alternative hypothesis H1 for (at least) every sufficiently small, positive t, since ψ(Pt,g ) = tP ψ˜P g + o(t) if ψ(P ) = 0. Thus the measures Ph/√n,g can be viewed as “local alternatives”. A test function φn is an estimator that takes is values in [0, 1]. The interpretation is that we reject the null hypothesis if the observed value of φn is 1, do not reject if it is 0, and reject it with probability φn (performing an additional experiment) if it is between 0 and 1. The following theorem shows that tests whose probabilities of the first kind (rejecting H0 if it is true) are bounded above by some level α necessarily have probabilities of the second kind (not rejecting H0 if it is false) bounded below by a certain Gaussian integral. Let zα = Φ−1 (1 − α) be the upper α-quantile of the standard normal distribution. Theorem 2.11. Let the functional ψ: P → R be differentiable at P relative to the tangent space P˙ P with efficient influence function ψ˜P . Suppose that ψ(P ) = 0. Then for every sequence of tests φn such that sup

Qn φn ≤ α ∈ (0, 1),

Q:ψ(Q)≤0

and every g ∈ P˙ P with P ψ˜P g > 0 and every h > 0,

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P ψ˜P g n√ . lim sup Ph/ n,g φn ≤ 1 − Φ zα − h n→∞ (P ψ˜2 )1/2 P

It is reasonable to expect that a test based on an efficient estimator is efficient as a test, and this is true, as we now show using the preceding theorem. Suppose that the sequence of estimators Tn is asymptotically efficient for ψ(P ) at P and that Sn is a consistent sequence of estimators of its variance P ψ˜P2 . Then the test that rejects H0 : ψ(P ) = 0 for √ asymptotic−1 nTn /Sn ≥ Φ (1 − α) attains the upper bound of the theorem. The critical value zα is chosen exactly √ so that the asymptotic probability of an error of the first kind is α: PP ( nTn /Sn ≥ zα ) → α. Lemma 2.12. Let the functional ψ: P → R be differentiable at P with ψ(P ) = 0. Suppose that the sequence Tn is regular at P with a N (0, P ψ˜P2 )P limit distribution. Furthermore, suppose that Sn2 → P ψ˜P2 . Then, for every ˙ h ≥ 0 and g ∈ PP , √nT

P ψ˜P g n . ≥ z α = 1 − Φ zα − h lim Ph/√n,g n→∞ Sn (P ψ˜P2 )1/2 Proof. By the efficiency of Tn and the differentiability of ψ, the sequence √ nTn converges under Ph/√n,g to a normal distribution with mean hP ψ˜P g

and variance P ψ˜P2 . Thus the lemma follows by simple algebra. Example 2.13 (Wilcoxon test). Suppose that the observations are two independent random samples X1 , . . . , Xn and Y1 , . . . , Yn from distribution functions F and G, respectively. To fit this two-sample problem in the present i.i.d. set-up, we pair the two examples and think of (Xi , Yi ) as a single observation from the
product measure F × G on R2 . We wish
to test 1the null hypothesis H0 : F dG ≤ 12 versus the alternative H : F dG > 2 . The Wilcoxon 1
test rejects H0 for large values of Fn dGn , where Fn and Gn are the empirical distribution functions of the two samples. This test is asymptotically efficient relative to the model in which F and G are completely unknown. This gives a different perspective on this test, which is usually presented as being asymptotically optimal for testing a difference of location in the logistic location-scale family. Actually, this finding is an example of the general principle that, in the situation that the underlying distribution of the observations is completely unknown, empirical-type statistics are asymptotically efficient for whatever they naturally estimate or test. The present conclusion concerning the Wilcoxon test extends to most other test statistics. By the preceding lemma, the efficiency of the test follows from the efficiency of the Wilcoxon statistic as an estimator for the function ψ(F × G) =
F dG. We do not give the complete argument for this, but note that it could be derived from the efficiency of the Fn for F and of Gn for G, which we noted in Example 2.10, either by applying a preservation theorem of efficiency, or by similar arguments.

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All three theorems presented in this section give a special role to normal distributions with covariance matrix P ψ˜P ψ˜PT . We have motivated the covariance matrix by the Cram´er-Rao theorem, but the normality is a new element. That “normal limit distributions are best” was proved for parametric models in the 1970s by H´ ajek, and is best explained from Le Cam’s theory of limiting experiments. This theory shows that the sequence of statistical experiments n√ ˙ (P1/ n,g : g ∈ PP )

converges in the weak sense of Le Cam to a Gaussian location experiment, indexed by the tangent set P˙ P . We do not discuss this convergence theory here, but do present a fourth theorem that is more in its spirit. Theorem 2.14. Suppose that Tn are estimators with values in a separable, Banach space D such that, for every g ∈ P˙ P and a probability measure Lg ,  √  n Tn − ψ(P1/√n,g )
Lg , under P1/√n,g . If ψ is differentiable at P , relative to P˙ P , then for any orthonormal sequence g1 , . . . , gm in L2 (P ) there exists a measurable map T : Rm × [0, 1] → D such that T − ψ˙ P (g) is distributed as Lg if the  law of T is calculated under the product of the normal measure with mean g, g1 P , . . . , g, gm P and covariance the identity and the uniform measure on [0, 1]. The measurable map T in this theorem should be regarded as a randomized estimator T = T (X, U ) in a statistical experiment that consists of observing a vector X = (X1 , . . . , Xm ) of m independent normal variables, with means gi , gP depending on an unknown parameter g and unit variance. The estimator is allowed to depend also on an auxiliary uniform variable U that can be generated by the statistician. (For many purposes it is not helpful to use randomization, but sometimes, as with nonconvex loss functions, it may be.) The theorem shows that asymptotically the problem of statistical inference about ψ(P1/√n,g ) based on a sample of size n from P1/√n,g , where g is unknown, is matched  by the problem of estimating ψ˙ P (g) based on X. Here we could restrict g = i=1 ai gi to the linear span of g1 , . . . , gm and develop m ˙ the parameter of interest ψ˙ P (g)  = i=1 ai ψP (gi ). Then we are to make inference about a linear function ai di based on a normal Nm (a, I)-distributed vector, which is a well-studied problem with simple solutions. The preceding theorems are merely specifications to particular problems (minimax estimation, equivariant estimation, or uniformly most powerful testing) of this Gaussian approximation. Using the preceding theorem we could obtain a load of other concrete statements on asymptotic lower bounds, provided we can solve the particular question in the Gaussian experiment. For instance, we can derive statements for tangent sets that do not satisfy the convexity or linearity requirements of the preceding theorems; we can consider loss functions that

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are not subconvex; or we can consider testing of higher-dimensional functionals. The problem with testing a parameter of dimension 2 or higher is that no uniformly most powerful, unbiased test does exist and hence an optimal test can only be defined through restricting the class of tests or working with envelope power functions. Appropriate restriction through invariance will of course lead to the same conclusion that tests best on best regular estimator sequences are best invariant tests. Rather than using finitely many functions g1 , . . . , gm , we could have used an infinite sequence g1 , g2 , . . . (unless L2 (P ) is finite dimensional). The analogous result will be true. However, the analysis of an infinite-dimensional Gaussian experiment will proceed by finite-dimensional approximation, so not much is gained by this formulation. We have a similar reservation against a representation of the Gaussian experiment using a Brownian motion with drift (as in [22]). It is impossible to perform direct calculations on risks of estimators which are measurable functions of Brownian motion and hence it will be necessary to approximate the experiment by finite-dimensional ones in any case. Proofs of generalizations of the preceding theorems are given in Lecture 4.

2.2 Efficient Score Functions A function ψ(P ) of particular interest is the parameter θ in a semiparametric model {Pθ,η : θ ∈ Θ, η ∈ H}. Here Θ is an open subset of Rk and H is an arbitrary set, typically of infinite dimension. The information bound for the functional of interest ψ(Pθ,η ) = θ can be conveniently expressed in an “efficient score function”. As submodels, we use paths of the form t → Pθ+ta,ηt , for given paths t → ηt in the parameter set H. The score functions for such submodels (if they exist) will typically have the form of a sum of “partial derivatives” with respect to θ and η. If ˙θ,η is the ordinary score function for θ in the model where η is fixed, then we expect ∂ log dPθ+ta,ηt = aT ˙θ,η + g. ∂t |t=0 The function g has the interpretation of a score function for η when θ is fixed, and will run through an infinite-dimensional set if we are concerned with a “true” semiparametric model. We refer to this set as the tangent set for η, and denote it by η P˙ Pθ,η . The parameter ψ(Pθ+ta,ηt ) = θ +ta is certainly differentiable with respect to t in the ordinary sense, but is, by definition, differentiable as a parameter on the model if and only if there exists a function ψ˜θ,η such that a=

∂ ψ(Pθ+ta,ηt ) = ψ˜θ,η , aT ˙θ,η + gPθ,η , ∂t |t=0

a ∈ Rk , g ∈ η P˙ Pθ,η .

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Setting a = 0, we see that ψ˜θ,η must be orthogonal to the tangent set η P˙ Pθ,η for the nuisance parameter. Define Πθ,η as the orthogonal projection onto the closure of the linear span of η P˙ Pθ,η in L2 (Pθ,η ). Definition 2.15. (i) The efficient score function for θ is ˜θ,η = ˙θ,η − Πθ,η ˙θ,η . (ii) The efficient information matrix for θ is I˜θ,η = Pθ,η ˜θ,η ˜Tθ,η . Lemma 2.16. Suppose that for every a ∈ Rk and every g ∈ η P˙ Pθ,η there exists a path t → ηt in H such that   1/2 1/2 dPθ+ta,ηt − dPθ,η t

1/2 − 12 (aT ˙θ,η + g) dPθ,η

2

→ 0.

(2.2)

If I˜θ,η is nonsingular, then the functional ψ(Pθ,η ) = θ is differentiable at Pθ,η relative to the tangent set P˙ Pθ,η = lin ˙θ,η + η P˙ Pθ,η with efficient influence function ψ˜θ,η = I˜−1 ˜θ,η . θ,η

Proof. The given set P˙ Pθ,η is a tangent set by assumption. The function ψ is differentiable with respect to this tangent set since −1 ˜ −1 ˜ θ,η , ˙Tθ,η Pθ,η a = a. θ,η , aT ˙θ,η + gPθ,η = I˜θ,η I˜θ,η

The last equality follows, because the inner product of a function and its orthogonal projection is equal to the square length of the projection. Thus, we may replace ˙θ,η by ˜θ,η .

Consequently, an estimator sequence is asymptotically efficient for estimating θ if n √ 1 ˜−1 ˜ I θ,η (Xi ) + oPθ,η (1). n(Tn − θ) = √ n i=1 θ,η This is very similar to the situation for efficient estimators in parametric models. The only difference is that the ordinary score function ˙θ,η is replaced by the efficient score function (and similarly for the informations). The intuitive explanation is that a part of the score function for θ can also be accounted for by score functions for the nuisance parameter η. When the nuisance parameter is unknown, a part of the information for θ is “lost”, and this corresponds to a “loss” of a part of the score function. Example 2.17 (Symmetric location). Suppose that the model consists of all densities x → η(x − θ) with θ ∈ R and the “shape” η symmetric about 0 with finite Fisher information for location Iη . Thus, the observations are sampled from a density that is symmetric about θ.   By the symmetry, the density can equivalently be written as η |x − θ| . It follows that any score function for the nuisance parameter η is necessarily a

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function of |x−θ|. This suggests a tangent set containing functions of the form   a(η  /η)(x − θ) + b |x − θ| . It is not hard to show that all square-integrable functions of this type with mean zero occur as score functions in the sense of (2.2). A symmetric density has an asymmetric derivative and hence an asymmetric score function for location. Therefore, for every b, Eθ,η

  η (X − θ) b |X − θ| = 0. η

Thus, the projection of the θ-score onto the set of nuisance scores is zero and hence the efficient score function coincides with the ordinary score function. This means that there is no difference in information about θ whether the form of the density is known or not known, as long as it is known to be symmetric. This surprising fact was discovered by Stein in 1956, and has been an important motivation in the early work on semiparametric models. Even more surprising is that the information calculation is not misleading. There exist estimator sequences for θ whose definition does not depend on η that have asymptotic variance Iη−1 under any true η! We shall see this in Lecture 8. Example 2.18 (Regression). Let gθ be a given set of functions indexed by a parameter θ ∈ Rk , and suppose that a typical observation (X, Y ) follows the regression model E(e| X) = 0. Y = gθ (X) + e, This model includes the logistic regression model, for gθ (x) = 1/(1 + e−θ x ). It is also a version of the ordinary linear regression model. However, in this example we do not assume that X and e are independent, but only the relations in the preceding display, apart from qualitative smoothness conditions that ensure existence of score functions, and the existence of moments. We shall write the formulas assuming that a density η. Thus, the  (X, e) possesses  observation (X, Y ) has a density η x, y − g (x) , where η is (essentially) only θ
restricted by the relations eη(x, e) de ≡ 0. Since
any perturbation ηt of η within the model must satisfy this same relation eηt (x, e) de = 0, it is clear that  score functions for the nuisance parameter η are functions a x, y − gθ (x) that satisfy
 ea(X, e) η(X, e) de
= 0. E ea(X, e)| X) = η(X, e) de T

By the same argument as for nonparametric models all square-integrable functions of  this type that have mean zero are score functions. Since the relation E ea(X, e)| X) = 0 is equivalent to the orthogonality in L2 (η) of a(x, e) to all functions of the form eh(x), it follows that the set of score functions for η is the orthocomplement of the set eH, of all functions of the  form (x, y) → y − gθ (x) h(x) within L2 (Pθ,η ), up to centering at mean zero.

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Thus, we obtain the efficient score function for θ by projecting the ordinary score function ˙θ,η (x, y) = −η2 /η(x, e)g˙ θ (x) onto eH. The projection of an arbitrary function b(x, e) onto the functions eH is a function eh0 (x) such that Eb(X, e)eh(X) = Eeh0 (X)eh(X) for all measurable functions h. This can be solved for h0 to find that the projection operator takes the form   E b(X, e)e| X . ΠeH b(X, e) = e E(e2 | X) This readily yields the efficient score function  
Y − gθ (X) g˙ θ (X) eg˙ θ (X) η2 (X, e)e de ˜
θ,η (X, Y ) = − = . E(e2 | X) E(e2 | X) η(X, e) de   The efficient information takes the form I˜θ,η = E g˙ θ g˙ θT (X)/E(e2 | X) . Notes The study of the symmetric location model has a long history. That the scores for the location parameter and the shape parameter were orthogonal was first noted by Stein in [34]. Several authors subsequently worked on defining adaptive estimators. A summary approach was given by Bickel in [2], which provided a starting point to extensions to more general models. The convolution and minimax theorems for parametric models are due to H´ ajek, see [10] and [11]. The semiparametric versions given here are, in a way, simple extensions of these theorems. The role of convexity or linearity of tangent spaces for these theorems was investigated in [36], which is also the basis of Theorem 2.15. Efficient score functions were presented by Begun, Hall, Huang and Wellner, in [1], as an alternative to the (more general) presentations by Levit and Pfanzagl.

3. Lecture: Calculus of Scores

In this lecture we introduce a “calculus of scores”, which is a useful way of finding efficient influence functions in models that are parametrized.

3.1 Score and Information Operators The method to find the efficient influence function of a parameter given in the preceding lecture is the most convenient method if the model can be naturally partitioned in the parameter of interest and a nuisance parameter. For many parameters such a partition is impossible, or, at least, unnatural. Furthermore, even in semiparametric models it can be worthwhile to derive a more concrete description of the tangent set for the nuisance parameter, in terms of a “score operator”. Consider first the situation that the model P = {Pη : η ∈ H} is indexed by a parameter η that is itself a probability measure on some measurable space. We are interested in estimating a parameter of the type ψ(Pη ) = χ(η) for a given function χ: H → Rk on the model H. The model H gives rise to a tangent set H˙ η at η. If the map η → Pη is differentiable in an appropriate sense, then its derivative will map every score b ∈ H˙ η into a score g for the model P. To make this precise, we assume that a smooth parametric submodel t → ηt induces a smooth parametric submodel t → Pηt , and that the score functions b of the submodel t → ηt and g of the submodel t → Pηt are related by g = Aη b. Then Aη H˙ η is a tangent set for the model P at Pη . Since Aη turns scores for the model H into scores for the model P it is called a score operator. Ahead it is seen that if η and Pη are the distributions of an unobservable Y and an observable X = m(Y ), respectively, then the score operator is a conditional expectation. More generally, it can be viewed as a derivative of the map η → Pη . We assume that Aη , as a map Aη : lin H˙ η ⊂ L2 (η) → L2 (Pη ), is continuous and linear. Next, assume that the function η → χ(η) is differentiable with influence function χ ˜η relative to the tangent set H˙ η . Then, by definition, the function

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ψ(Pη ) = χ(η) is pathwise differentiable relative to the tangent set P˙ Pη = Aη H˙ η if and only if there exists a vector-valued function ψ˜P such that η

ψ˜Pη , Aη bPη =

∂ ∂ ψ(Pηt ) = χ(ηt ) = χ ˜η , bη , ∂t |t=0 ∂t |t=0

b ∈ H˙ η .

This equation can be rewritten in terms of the adjoint score operator A∗η : L2 (Pη ) → lin H˙ η . By definition this satisfies h, Aη bPη = A∗η h, bη for every h ∈ L2 (Pη ) and b ∈ H˙ η . Note that we define A∗η to have range lin H˙ η , so that it is the adjoint of Aη : H˙ η → L2 (Pη ). This is the adjoint of an extension Aη : L2 (η) → L2 (Pη ) followed by the orthogonal projection onto lin H˙ η . Fact 3.1. Every continuous, linear map A: H1 → H2 between two Hilbert spaces has an adjoint map A∗ : H2 → H1 , which is a continuous, linear map that satisfies and is uniquely determined by the equations A∗ h2 , h1 1 = ˜1 h2 , Ah1 2 for every hi ∈ Hi . If A is considered the restriction to H1 ⊂ H ˜ ˜ of a continuous, linear map A: H1 → H2 with domain a Hilbert space that ˜ 1 → H1 the orthogonal contains H1 isometrically, then A∗ = Π A˜∗ for Π: H ˜ projection of H1 onto H1 . The preceding display is equivalent to ˜η . A∗η ψ˜Pη = χ

(3.1)

We conclude that the function ψ(Pη ) = χ(η) is differentiable relative to the tangent set P˙ Pη = Aη H˙ η if and only if this equation can be solved for ψ˜Pη ; equivalently, if and only if χ˜η is contained in the range of the adjoint A∗η . Since A∗η is not necessarily onto lin H˙ η , not even when it is one-to-one, this is a condition! For multivariate functionals equation (3.1) is to be understood coordinatewise. Two solutions ψ˜Pη of (3.1) can differ only by an element of the kernel N(A∗η ) of A∗η , which is the orthocomplement R(Aη )⊥ of the range of Aη : lin H˙ η → L2 (Pη ). Thus, there is at most one solution ψ˜Pη that is contained in R(Aη ) = lin Aη H˙ η , the closure of the range of Aη , as required. If χ ˜η is contained in the smaller range of A∗η Aη , then equation (3.1) can be solved, of course, and the solution can be written in the attractive form ψ˜Pη = Aη (A∗η Aη )− χ ˜η .

(3.2)

Here A∗η Aη is called the information operator, and (A∗η Aη )− is a “generalized ˜η is a solution inverse”. (Here this will not mean more than that b = (A∗η Aη )− χ ˜η .) The following lemma shows that this attractive to the equation A∗η Aη b = χ form is available for any functional χ if the range of the score operator is closed, a situation which unfortunately fails often.

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Fact 3.2. Let A: H1 → H2 be a continuous linear map between two Hilbert spaces. Then equivalent are: (i) (ii) (iii) (iv)

R(A) is closed. R(A∗ ) is closed. R(A∗ A) is closed. R(A∗ A) = R(A∗ ).

Fact 3.3. Let A: H1 → H2 be a continuous linear map between two Hilbert spaces. Then (i) N(A) = R(A∗ )⊥ . (ii) N(A∗ ) = R(A)⊥ . Furthermore, the map A∗ A: H1 → H1 is one-to-one, onto and has a continuous inverse if and only if A is one-to-one and R(A) is closed if and only if A∗ A is one-to-one and onto. So far we have assumed that the parameter η is a probability distribution, but this is not necessary. Consider the more general situation of a model P = {Pη : η ∈ H} indexed by a parameter η running through an arbitrary set H. Let Hη be a subset of a Hilbert space that indexes “directions” b in which η can be approximated within H. Suppose that there exist continuous, linear operators Aη : lin Hη → L2 (Pη ) and χ˙ η : lin Hη → Rk , and for every b ∈ Hη a path t → ηt such that, as t ↓ 0,   1/2 1/2 2 dPηt − dPη − 12 Aη b dPη1/2 → 0, t χ(ηt ) − χ(η) → χ˙ η b. t By the Riesz representation theorem for Hilbert spaces, the “derivative” χ˙ η has a representation as an inner product χ˙ η b = χ ˜η , bHη for an element χ ˜η ∈ lin Hkη . The preceding discussion can be extended to this abstract setup. Theorem 3.4. The map ψ: P → Rk given by ψ(Pη ) = χ(η) is differentiable at Pη relative to the tangent set Aη Hη if and only if each coordinate function of χ ˜η is contained in the range of A∗η : L2 (Pη ) → lin Hη . The efficient influence ˜η is contained in function ψ˜Pη satisfies (3.1). If each coordinate function of χ the range of A∗η Aη : lin Hη → lin Hη , then it also satisfies (3.2). Proof. By assumption, the set Aη Hη is a tangent set. The map ψ is differentiable relative to this tangent set (and the corresponding submodels t → Pηt ) by the argument leading up to (3.1).

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The condition (3.1) is odd. By definition, the influence function χ˜η is contained in the closed linear span of Hη and the operator A∗η maps L2 (Pη ) into lin Hη . Therefore, the condition is certainly satisfied if A∗η is onto. There are two reasons why it may fail to be onto. First, its range R(A∗η ) may be a proper subspace of lin Hη . Since b ⊥ R(A∗η ) if and only if b ∈ N(Aη ), this can happen only if Aη is not one-to-one. This means that two different directions b may lead to the same score function Aη b, so that the information matrix for the corresponding two-dimensional submodel is singular. A rough interpretation is that the parameter is not locally identifiable and it is not surprising that we have a problem. Second, the range space R(A∗η ) may be dense, but not closed. Then for any χ ˜η there exist elements in R(A∗η ) that are arbitrarily close to χ ˜η , but (3.1) may still fail. This is harder to understand, but it happens quite often. The following theorem shows that failure has serious consequences. / R(A∗η ), then Theorem 3.5. In the above setting, if χ ˜η ∈ (i) there exists no estimator sequence for χ(η) that is regular at Pη . (ii) χ ˜η , b2η sup 2 = ∞. b∈Hη Aη bPη Proof. We shall only give the proof of (ii). (See [38] for a proof of (i).) The proof of (ii) can be carried out using the spectral decomposition of the information operator and spectral calculus. (See for instance [33] for this background.) For simplicity of notation, we drop the index
η throughout the proof. The spectral decomposition takes the form A∗ A = λ dPλ for λ → Pλ the spectral resolution of the nonnegative, self-adjoint operator A∗ A. (In simple cases, the formal integral is a sum over the (countable many, nonnegative) eigenvalues of A∗ A and the Pλ are orthogonal projections on the corresponding eigenspaces. In general, the spectral resolution may be continuous.) Next the operator  √ λ dPλ (A∗ A)1/2 = is a square root in that it is nonnegative, self-adjoint and has A∗ A as its square. The adjoint A∗ can be expressed in this square root through the polar decomposition A∗ = (A∗ A)1/2 U , for U : L2 (P ) → R((A∗ A)1/2 ) = N((A∗ A)1/2 )⊥ an operator whose restriction to R(A) is an isometry and has R(A)⊥ as its kernel. It follows that the ranges of A∗ and (A∗ A)1/2 are identical.
The spectral calculus also gives a meaning to integrals of the type define operators, which f (λ) dPλ for general functions f . Such


expressions can be manipulated with rules such as f (λ) dPλ g(λ) dPλ = f (λ)g(λ) dPλ . Furthermore, to every b ∈ H corresponds a spectral measure µb , which is a measure on the interval [0, A∗ A] containing the spectrum with the property

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that  f (λ) dPλ b, b = |f |2 dµb for every function f that is well-behaved on the spectrum of A∗ A. We then obtain that      R(A∗ ) = R (A∗ A)1/2 = b: λ−1 dµb (λ) < ∞ . Therefore, if χ ˜ is not contained in the range of A∗ , then we must have at least one of  1λ>0 λ−1 dµχ˜ (λ) = ∞. µχ˜ {0} > 0, or
In the first case we evaluate the quotient in (ii) at b = P{0} χ ˜ = {0} dPλ χ. ˜
∗ ˜ =
For this choice we have by spectral calculus that A Ab = λ dPλ P{0} χ ˜ = 0, whereas χ, ˜ b = µχ˜ {0} > 0. This yields the quotient (> 1{0} (λ)λ dPλ χ display is 0)2 /0 in (ii). In the case that the second possibility in the preceding
˜ valid we evaluate the quotient in (ii) at the sequence bn =
1λ≥1/n λ−1 dPλ χ. ∗ Ab = 1 dP χ ˜ → For this choice we have by spectral calculus that A n λ λ≥1/n
χ, ˜ whereas χ, ˜ bn  = 1λ≥1/n λ−1 dµχ˜ (λ) → ∞. Thus the quotient in (ii) is infinite.

3.1.1 Information Loss Models Suppose that a typical observation is distributed as a measurable transformation X = m(Y ) of an unobservable variable Y . Assume that the form of m is known and that the distribution η of Y is known to belong to a class H. This yields a natural parametrization of the distribution Pη of X. A nice property of differentiability in quadratic mean is that it is preserved under “censoring” mechanisms of this type: if t → ηt is a differentiable submodel of H, then the induced submodel t → Pηt is a differentiable submodel of {Pη : η ∈ H}. Furthermore, the score function g = Aη b (at t = 0) for the induced model t → Pηt can be obtained from the score function b (at t = 0) of the model t → ηt by taking a conditional expectation:   Aη b(x) = Eη b(Y )| X = x . If we consider the scores b and g as the carriers of information about t in the variables Y ∼ ηt and X ∼ Pηt , respectively, then the intuitive meaning of the conditional expectation operator is clear. The information contained in the observation X is the information contained in Y diluted (and reduced) through conditioning. Lemma 3.6. Suppose that {ηt : 0 < t < 1} is a collection of probability measures on a measurable space (Y, B) such that for some measurable function b: Y → R   1/2 2 dηt − dη 1/2 − 12 b dη 1/2 → 0. t

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For a measurable map m: Y → X let Pη be the distribution of m(Y ) if Y has law η and let Aη b(x) be the conditional expectation of b(Y ) given m(Y ) = x. Then   1/2 1/2 2 dPηt − dPη − 12 Aη b dPη1/2 → 0. t Proof. For simplicity of notation we assume that the measures ηt and η have densities ht and h relative to a fixed probability measure ν. (If this is not the case, choose ν = νt = 12 (ηt + η) dependent on t and add t’s throughout the following.) Furthermore, we assume that b is uniformly bounded by M . (If this is not the case truncate b at Mt → ∞ and add t’s in the following.) Then νu2t → 0 for 1/2 h − h1/2 ut = t − 12 bh1/2 . t Define µ to be the law of X = m(Y ) if Y is distributed according to ν. Then     pt (x) = Eν ht (Y )| X = x , and p(x) = Eν h(Y )| X = x are densities of Pηt and Pη with respect to µ. In case of the second one, this follows from the equations      Pη (A) = 1A m(y) dη(y) = Eν 1A m(Y ) h(Y )    = Eν 1A (X)Eν h(Y )| X = p(x) dµ(x). A

By a similar argument, we have, almost surely under Pη ,   Aη b(X) p(X) = Eν b(Y )h(Y )| X . From the definition of ut we obtain that ht = h + tbh + t2 u2t + t(tut bh1/2 + 2ut h1/2 + 14 tb2 h). Evaluating these functions at Y and taking conditional expectations with respect to X, we find pt = p + t(Aη b) p + c + d, where c and d satisfy   c(X) = t2 Eν u2t (Y )| X ,

2   |d(X)|2 = t2 Eν tut bh1/2 + 2ut h1/2 + 14 tb2 h (Y )| X

2    t2 Eν ut h1/2 (tM + 1) + tM 2 h (Y )| X

   t2 Eν u2t (Y )| X (tM + 1)2 + t2 M 4 p(X) p(X),

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by the Cauchy–Schwarz inequality. By a Taylor expansion (see Lemma 3.7), we conclude that on the set A = {p > 0}  p1/2 − p1/2 t

t

− 12 (Aη b)p1/2

2

   Eν u2t (Y )| X (tM + 1)2 + t2 M 4 p(X) 2    1 + Eν u2t (Y )| X +  √ − 1 M 2 p(X). 1 − Mt

The integral over the set A of this function relative to µ converges to zero as t → 0.   the equation η m−1 (Ac ) = Pη (Ac ) = 0 implies that Pηt (Ac ) =  Finally,  ηt m−1 (Ac ) = o(t2 ), because ν(tu2t 1B ) = ηt (B) if η(B) = 0. Thus the integral of the preceding display over the set Ac converges to zero as well.

Lemma 3.7. For any real numbers a, b, c, d with a > 0, b/a ≤ ε < 1, c ≥ 0 and a + b + c + d ≥ 0  2 b2  1 √ √ b 2 3d2    − 1 + 3c +  √ .  a + b + c + d − a − 12 √  ≤ a(1 − ε) a a 1−ε If we consider Aη as an operator Aη : L2 (η) → L2 (Pη ), then its adjoint A∗η : L2 (Pη ) → L2 (η) is a conditional expectation operator also, reversing the roles of X and Y ,   A∗η g(y) = Eη g(X)| Y = y . This follows since, by the usual rules for conditional expectations,     EE g(X)| Y b(Y ) = Eg(X)b(Y ) = Eg(X)E b(Y )| X . In the “calculus of scores” of Theorem 3.4 the adjoint is understood to be the adjoint of Aη : Hη → L2 (Pη ) and hence to have range lin Hη ⊂ L2 (η). Then the conditional expectation in the preceding display needs to be followed by the orthogonal projection onto lin Hη . Example 3.8 (Mixtures). Suppose that a typical observation X possesses a conditional density p(x| z) given an unobservable variable Z = z. If the unobservable Z possesses an unknown probability distribution η, then the observations are a random sample from the mixture density  pη (x) = p(x| z) dη(z). This is a missing data problem if we think of X as a function of the pair Y = (X, Z). A score for the mixing distribution η in the model for Y is a function b(z). Thus, a score space for the mixing distribution in the model for X consists of the functions

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 Aη b(x) = Eη b(Z)| X = x =




b(z) p(x| z) dη(z)
. p(x| z) dη(z)

If the mixing distribution is completely unknown, which we assume, then the tangent set H˙ η for η can be taken equal to the maximal tangent set {b ∈ L2 (η): ηb = 0}. In particular, consider the situation that the p(x| z) belongs to an  kernel  exponential family, i.e. p(x| z) = c(z)d(x) exp z T x . Mixtures over exponential families of this type give relatively large models. In fact, if the interior of the support of η is nonempty, than the tangent set Aη H˙ η is dense in the maximal tangent set {g ∈ L2 (Pη ): Pη g = 0}. We show this below. This has as a consequence that empirical estimators Pn g, for a fixed squared-integrable function g, are efficient estimators for the functional ψ(η) = Pη g. For instance, the sample mean is asymptotically efficient for estimating the mean of the observations. This is somewhat surprising, because the mixture densities may still possess very special properties. For instance, mixtures over the exponential scale family p(x| z) = zezx 1x>0 are monotone densities, and mixtures over the normal location family are extremely smooth. In terms of entropy the second collection of mixtures is almost finite-dimensional and there exist estimators pηˆ that √ obtain a rate of convergence in the Hellinger distance of the order log n/ n. Thus the set of all exponential mixtures can be far from being equal to the nonparametric model. The closure of the range of the operator Aη is the orthocomplement of the kernel N(A∗η ) of its adjoint. Hence our claim is proved if this kernel is zero. The equation    0 = A∗η g(z) = E g(X)| Z = z = g(x) p(x| z) dx says exactly that g(X) is a zero-estimator under p(x| z). Since the adjoint is defined on L2 (η), the equation 0 = A∗η g should be taken to mean A∗η g(Z) = 0 almost surely under η. In other words, the display is valid for every z in a set of η-measure 1. If the support of η contains a limit point, then this set is rich enough to conclude that g = 0, by the completeness of the exponential family. The same argument shows also that the range of the score operator, equivalently the range of its adjoint, is not closed in this example. This has √ as a consequence that many functionals y → χ(η) are not in the realm of the ntheory of estimation. As an example consider the functional χ(η) = η(A) for a given set A. This has influence function χ ˜ = 1A − η(A), which is contained in the range of A∗η if and only if there exists a measurable function g such that  T 1A (z) = g(x)c(z)d(x)ez x dµ(x), η-a.e.. The completeness of the exponential family shows that the A must have probability 0 or 1 under η. Functionals as these belong to the realm of inverse

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problems. Not much is known about them today. The deconvolution problem (i.e. p(x| z) a location family) has best been studied, with a characterization of rates for estimating the mixing distribution function and its derivatives using Fourier inversion methods. Even in this case very little is known concerning standard methods of estimation, such as maximum likelihood. If the support of η does not contain a limit point, then the preceding approach to show that the tangent set is dense fails. However, we may reach almost the same conclusion by using a different type of scores. The paths ηt = (1 − ta)η + taη1 are well-defined for 0 ≤ at ≤ 1, for any fixed a ≥ 0 and η1 , and lead to scores p

∂ η1 log pηt (x) = a (x) − 1 . ∂t |t=0 pη This is certainly a score in a pointwise sense, and can be shown to be a score in the L2 -sense provided that it is in L2 (Pη ). If g ∈ L2 (Pη ) has Pη g = 0 and is orthogonal to all scores of this type, then p

η1 − 1 , every η1 . 0 = Pη1 g = Pη g pη If the set of distributions {Pη : η ∈ H} is complete, then we can typically conclude that g = 0 almost surely. Then the closed linear span of the tangent set is equal to the nonparametric, maximal tangent set. Since this set of scores is also a convex cone, Theorems 2.7 and 2.5 next show that nonparametric estimators are asymptotically efficient.

3.2 Semiparametric Models In a semiparametric model {Pθ,η : θ ∈ Θ, η ∈ H}, the pair (θ, η) plays the role of the single η in the preceding general discussion. The two parameters can be perturbed independently, and the score operator can be expected to take the form Aθ,η (a, b) = aT ˙θ,η + Bθ,η b. Here Bθ,η : Hη → L2 (Pθ,η ) is the “score operator” for the nuisance parameter. The domain of the operator Aθ,η : Rk × lin Hη → L2 (Pθ,η ) is a Hilbert space relative to the inner product (a, b), (α, β)η = aT α + b, βHη . Thus this example fits in the general set-up, with Rk × Hη playing the role of the earlier Hη . We shall derive expressions for the efficient influence functions of θ and η. Fact 3.9. Given a continuous, linear map A: H1 → H2 between Hilbert spaces, the operator A(A∗ A)−1 A∗ (if it exists) is the orthogonal projection of H1 onto the range space of A.

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The efficient influence function for estimating θ is expressed in the efficient score function for θ in Lemma 2.16, which is defined as the ordinary score function minus its projection onto the score-space for η. Presently, the ∗ latter space is the range of the operator Bθ,η . If the operator Bθ,η Bθ,η is continuously invertible (but in many examples it is not), then the operator ∗ ∗ Bθ,η )−1 Bθ,η is the orthogonal projection onto the nuisance score Bθ,η (Bθ,η space, and  ∗ ∗ Bθ,η )−1 Bθ,η )˙θ,η . (3.3) ˜θ,η = I − Bθ,η (Bθ,η ∗ ∗ ˙ This means that b = −(Bθ,η Bθ,η )−1 Bθ,η θ,η is a “least favourable direction” in H, for estimating θ. If θ is one-dimensional, then the submodel t → Pθ+t,ηt where ηt approaches η in this direction, has the least information for estimating t and score function ˜θ,η , at t = 0. A function χ(η) of the nuisance parameter can, despite the name, also be of interest. The efficient influence function for this parameter can be found from (3.1). The adjoint of Aθ,η : Rk × Hη → L2 (Pθ,η ), and the corresponding information operator A∗θ,η Aθ,η : Rk × Hη → Rk × lin Hη are given by, with ∗ : L2 (Pθ,η → lin Hη the adjoint of Bθ,η , Bθ,η

  ∗ g , A∗θ,η g = Pθ,η g ˙θ,η , Bθ,η    Iθ,η Pθ,η ˙θ,η Bθ,η · a A∗θ,η Aθ,η (a, b) = . ∗ ˙T ∗ b Bθ,η Bθ,η θ,η Bθ,η The diagonal elements in the matrix are the information operators for the parameters θ and η, respectively, the former being just the ordinary Fisher information matrix Iθ,η for θ. If η → χ(η) is differentiable as before, then the function (θ, η) → χ(η) is differentiable with influence function (0, χ ˜η ). Thus, for a real parameter χ(η), equation (3.1) becomes Pθ,η ψ˜Pθ,η ˙θ,η = 0,

∗ ˜ Bθ,η ˜η . ψPθ,η = χ

∗ ˜η is contained in the range of Bθ,η Bθ,η , then the If I˜θ,η is invertible and χ solution ψ˜Pθ,η of these equations is −1 ˜ ∗ ∗ Bθ,η (Bθ,η Bθ,η )− χ ˜η − Bθ,η (Bθ,η Bθ,η )− χ ˜η , ˙θ,η TPθ,η I˜θ,η θ,η .

The second part of this function is the part of the efficient score function for χ(η) that is “lost” due to the fact that θ is unknown. Since it is orthogonal to the first part, it adds a positive contribution to the variance. Example 3.10 (Cox model). We illustrate the general formulas by explicit calculations for the Cox model. This model is appropriate for this purpose, because the information operator can be obtained in a simple form, whereas in other models not much progress can be made beyond writing out formulas for the score operator and its adjoint.

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For later reference we consider the Cox model under right censoring. In this model we observe a random sample from the distribution of the variable X = (T ∧ C, 1{T ≤ C}, Z), where, given Z, the variables T and C are independent, as in the right censoring model, and (Z, T ) follows the Cox model. Thus, the density of X = (Y, ∆, Z) is given by

1−δ   δ −eθz Λ(y) θz e fC|Z (y| z) pZ (z). eθz λ(y)e−e Λ(y) 1 − FC|Z (y−| z) We make a number of assumptions, whose main purpose is to simplify the formulas and to ensure the existence of the inverse of the information operator. First, we assume that the covariate Z is bounded, and that the true conditional distribution of T given Z possesses a continuous Lebesgue density. Second, we assume that there exists a finite number τ > 0 such that P(C ≥ τ ) = P(C = τ ) > 0 and Pθ0 ,Λ0 (T > τ ) > 0. The latter condition is not unnatural: it is satisfied if the survival study is stopped at some time τ at which a positive fraction of individuals is still “at risk” (alive). Third, we assume that, for any measurable function h, the probability that Z = h(Y ) is positive. The function Λ now matters only on [0, τ ]; we shall identify Λ with its restriction to this interval. The score function for θ takes the form, with x = (y, δ, z) ˙θ,Λ (x) = δz − zeθz Λ(y). For any bounded, measurable function a: [0, τ ] → R, the path defined by dΛt = (1 + ta) dΛ defines a submodel passing through Λ at t = 0. Its score function at t = 0 takes the form  θz a dΛ. Bθ,Λ a(x) = δa(y) − e [0,y]

For unbounded functions a we could employ paths of the form dΛt = χ(ta) dΛ and obtain a score of the same form. The score operator can be viewed as an operator Bθ,Λ : L2 (Λ) → L2 (Pθ,Λ ), so we can take HΛ = L2 (Λ) or take HΛ equal to the subset of all bounded functions in L2 (Λ). ∗ of Bθ,Λ : L2 (Λ) → L2 (Pθ,Λ ), we To find a formula for the adjoint Bθ,Λ write ∗ g, a = g, Bθ,Λ aPθ,Λ Bθ,Λ   y

θz θz = EZ g(y, 1, z) a(y) − e a dΛ eθz e−e Λ(y) 0   × 1 − FC (y−| z) dΛ(y)   y

θz + EZ g(y, 0, z) −eθz a dΛ eθz e−e Λ(y) dFC (y| z). 0

Next we use Fubini’s theorem to
change the order of integration in the two

terms, rewriting the right side as a · · · ] dΛ. By definition the term appear∗ g. It is given by ing inside the square brackets is then Bθ,Λ

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  θz ∗ Bθ,Λ g(y) = EZ g(y, 1, z)eθz e−e Λ(y) 1 − FC (y−| z)    θz − EZ g(s, 1, z)e2θz 1y≤s e−e Λ(s) 1 − FC (s−| z) dΛ(s)  θz − EZ g(y, 0, z)e2θz e−e Λ(s) 1y≤s dFC (s| z). This is not the simple formula promised in the introduction, though it has the ∗ benefit of being obtainable by simple mechanical manipulations. Now, Bθ,Λ g for an arbitrary function g is not really what interests us: rather we would like ∗ ∗ ˙ Bθ,Λ g and for Bθ,Λ θ,Λ . to obtain formulas for the information operator Bθ,Λ For this we can continue our mechanical work by combining the formulas obtained so far. This is straightforward again, and for most examples this would be the end of the story. The Cox model is special in that clever partial integrations can next simply the formulas considerably. We shall not pursue this approach, as it is tedious and not insightful. Rather we obtain the desired formulas using a statistical principle: minus the mean of the observed information is the Fisher information. A preciser formulation of this principle is that, given probability densities x → ps,t (x) that depend smoothly on a parameter (s, t) ∈ R2 , we have Es,t

∂

∂

∂2 log ps,t log ps,t = −Es,t log ps,t . ∂s ∂t ∂s∂t

We apply this to the submodels (s, t) → Pθ,Λs,t for dΛs,t = (1 + sa + tb + stab) dΛ = (1 + sa) dΛ0,t at (s, t) = (0, 0). This gives ∂2 ∂ pθ,Λs,t = −Eθ,Λ Bθ,Λ0,t a ∂s∂t |s,t=0 ∂t |t=0  ab dΛ = Eθ,Λ eθZ [0,Y ]  = b(s)Eθ,Λ eθZ 1s≤Y a(s) dΛ(s).

Eθ,Λ (Bθ,Λ a)(Bθ,Λ b) = −Eθ,Λ

By definition of the adjoint, the left side of this display is also equal to ∗ the inner product of b and Bθ,Λ Bθ,Λ a in L2 (Λ). Thus we read off that the information operator is the multiplication operator given by

∗ Bθ,Λ Bθ,Λ a(s) = Eθ,Λ eθZ 1s≤Y a(s). ∗ ˙ The function Bθ,Λ θ,Λ can be obtained by a similar argument, using the submodel (s, t) → Pθ+s,Λt with dΛt = (1 + tb)dΛ. It is given by ∗ ˙ θ,Λ (s) = Eθ,Λ 1s≤Y ZeθZ . Bθ,Λ

It is remarkable that the information operator is already in its spectral form. It is a theorem in Hilbert space theory that every self-adjoint operator can

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be written as a multiplication operator, relative to an appropriate coordinate system. In the present case the information operator already takes the form of a multiplication operator relative to the original coordinate system. It is easy to invert a multiplication operator. In the present situation, if (θ, Λ) is a pair of parameters that satisfies the assumptions we have made, the multiplier function y → Eθ,Λ 1y≤Y eθZ is bounded away from zero on [0, τ ]. Thus the inverse of the information operator exists as a continuous operator and is given by

−1 ∗ (Bθ,Λ Bθ,Λ )−1 a(s) = Eθ,Λ eθZ 1s≤Y a(s). The efficient score function takes the general form ((3.3)), which, with the functions Li,θ (y) = E1Y ≥y Z i eθZ , reduces to  L1,θ L1,θ θz ˜ (y) − e (t) dΛ(t). θ,Λ (x) = δ z − z− L0,θ L0,θ [0,y] The efficient information for θ can be computed from this as  L1,θ 2 θZ ˜ Iθ,Λ = Ee (y) Gθ,Λ (y| Z) dΛ(y), Z− L0,θ where G(y| Z) = P(Y ≥ y| Z). This is strictly positive by the assumption that Z is not almost surely equal to a function of Y . The formula for I˜θ,Λ can be obtained by direct (but tedious, if not difficult) computations. Alternatively, we can use martingale theory. The process  1s≤T eθZ dΛ(s) Mt = 1T ≤t − [0,t]

is a martingale relative to the filtration generated by (Z, C) and 1T ≤s for s ≤ t, with predictable quadratic variation process  1s≤T eθZ dΛ(s). Mt  = [0,t]

(We have assumed that Λ is continuous.) The efficient score can be written as the integral  L1,θ (t) dMs . ˜θ,Λ (X) = 1t≤C Z − L0,θ Because the integrand is predictable, the integral can be viewed as both an ordinary Stieltjes integral and a stochastic integral. By the second interpretation we have that  L1,θ 2 2 ˜ Eθ,Λ (X) = E 1s≤C Z − (t) dMt . L0,θ This can be seen to reduce to the formula obtained previously. Notes This lecture is based on the papers [1] and [37].

4. Lecture: Gaussian Approximations

In this lecture we give proofs of the lower bound theorems stated in Lecture 2, in a more general setting. For completeness we start by a crash course on contiguity.

4.1 Contiguity Suppose we are given two probability measures P and Q on a measurable space (Ω, U), with densities p and q relative to some measure µ. We denote by dQ/dP the ratio q/p, which is with P -probability one well-defined and not depending on µ. In fact, it is a density of the absolutely continuous part of Q relative to P . (Note that we do no write dQa /dP and we do not assume that Q  P .) Let X: Ω → D be a measurable map in a metric space. Then (X, dQ/dP ) is a measurable map into D × R, and it induces a law L on this space if we equip (Ω, U) with P . If Q  P , then this law determines the law of X under Q, because in this case  dQ = v dL(x, v). Q(X ∈ B) = EP 1B (X) dP B×R The validity of this formula depends essentially on the absolute continuity of Q with respect to P , because a part of Q that is orthogonal with respect to P cannot be recovered from any P -law. Consider an asymptotic version of the problem. Let (Ωn , An ) be measurable spaces, each equipped with a pair of probability measures Pn and Qn . Under what conditions can a Qn -limit law of random vectors Xn : Ωn → Rk be obtained from suitable Pn -limit laws? In view of the above it is necessary that Qn is “asymptotically absolutely continuous” with respect to Pn in a suitable sense. The right concept is contiguity. Definition 4.1. The sequence Qn is contiguous with respect to the sequence Pn if Pn (An ) → 0 implies Qn (An ) → 0 for every sequence of measurable sets An . This is denoted Qn  Pn . The sequences Pn and Qn are mutually contiguous if both Pn  Qn and Qn  Pn . This is denoted Pn   Qn .

4.1 Contiguity

371

The name “contiguous” is standard, but perhaps conveys a wrong image. “Contiguity” suggests sequences of probability measures living next to each other, while the correct image is “on top of each other” (in the limit). Before answering the question of interest, we give two characterizations of contiguity in terms of the asymptotic behaviour of the likelihood ratios of Pn and Qn . The likelihood ratios dQn /dPn and dPn /dQn are nonnegative and satisfy dQn dPn EPn ≤ 1 and EQn ≤ 1. dPn dQn Thus, the sequences of likelihood ratios dQn /dPn and dPn /dQn are uniformly tight under Pn and Qn , respectively. By Prohorov’s theorem, every subsequence has a further weakly converging subsequence. The next lemma shows that the properties of the limit points determine contiguity. Lemma 4.2 (Le Cam’s first lemma). Let Pn and Qn be sequences of probability measures on measurable spaces (Ωn , An ). Then the following statements are equivalent: (i) (ii) (iii) (iv)

Qn  Pn ; n if dPn /dQn Q
U along a subsequence, then P(U > 0) = 1; Pn if dQn /dPn
V along a subsequence, then EV = 1; Pn n for any statistics Tn : Ωn → Rk : if Tn → 0, then Tn Q → 0.

We do not include a proof of this lemma, but note that the lemma is easy if the sequences Pn and Qn are constant. If (Ωn , Un ) = (Ω, U), Pn = P and Qn = Q for each n, then contiguity is equivalent to absolute continuity, and the lemma reduces to the equivalence of the three statements

dP dQ = 0 = 0, EP = 1. Q  P, Q dQ dP The lemma shows that these equivalences persist if the three statements are replaced by their asymptotic counterparts. n√ According to Lemma 1.9 the likelihood ratios of the measures P1/ and n n P for a given differentiable path t → Pt are asymptotically log-normally distributed with mean − 12 P g 2 and variance P g 2 . This makes these sequences of measures mutually contiguous. Example 4.3 (Asymptotic log normality). Let Pn and Qn be probability measures on arbitrary measurable spaces such that dPn Qn N (µ,σ2 )
e . dQn Then Qn  Pn . Furthermore, Qn   Pn if and only if µ = − 12 σ 2 . Since the (log normal) variable on the right is positive, the first assertion is immediate from (ii) of the theorem. The second follows from (iii) with the roles of Pn and Qn switched, upon noting that E exp N (µ, σ 2 ) = 1 if and only if µ = − 12 σ 2 .

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The following theorem solves the problem of obtaining a Qn -limit law from a Pn -limit law that we posed in the introduction. The result, a version of Le Cam’s third lemma, is in perfect analogy with the nonasymptotic situation. Theorem 4.4. Let Pn and Qn be sequences of probability measures on measurable spaces (Ωn , An ), and let Xn : Ωn → Rk be a sequence of maps. Suppose that Qn  Pn and dQn Pn
(X, V ). Xn , dPn n Then L(B) = E1B (X) V defines a probability measure, and Xn Q
L.

Proof. Since V ≥ 0, it follows with the help of the monotone convergence theorem that L defines a measure. By contiguity, EV = 1 and hence
L is a probability measure. It is immediate from the definition of L that f dL = Ef (X) V for every measurable indicator function f . Conclude, in steps, that the same is true for every simple function f , any nonnegative measurable function, and every integrable function. If f is continuous and nonnegative, then so is the function (x, v) → f (x) v on Rk × [0, ∞). Thus  dQn dPn ≥ Ef (X)V, lim inf EQn ,∗ f (Xn ) ≥ lim inf f (Xn ) dPn ∗ by the portmanteau lemma. Apply the portmanteau lemma in the converse n direction to conclude the proof that Xn Q


L. Example 4.5 (Le Cam’s third lemma). The name Le Cam’s third lemma is often reserved for the following result. If       dQn Pn µ Σ τ
Nk+1 , , Xn , log τ T σ2 − 12 σ 2 dPn then

Qn

Xn
Nk (µ + τ, Σ). In this situation the asymptotic covariance matrices of the sequence Xn are the same under Pn and Qn , but the mean vectors differ by the asymptotic covariance τ between Xn and the log likelihood ratios. The statement is a special case of the preceding theorem. Let (X, W ) have the given (k+1)-dimensional normal distribution. By the continuous mapping theorem, the sequence (Xn , dQn /dPn ) converges in distribution under Pn to (X, eW ). Since W is N (− 12 σ 2 , σ 2 )-distributed, the sequences Pn and Qn are mutually contiguous. According to the abstract version of Le Cam’s third n lemma, Xn Q L with L(B) = E1B (X)eW . The characteristic function of L
itT x
T is e dL(x) = Eeit X eW . This is the characteristic function of the given normal distribution at the vector (t, −i). Thus

4.2 Gaussian Representations



T

eit

x

T

1 1 µ− 2 σ 2 − 2 (tT ,−i)

T

1 (µ+τ )− 2 tT Σt

dL(x) = eit = eit



Σ τ τ T σ2



373



t −i

.

The right side is the characteristic function of the Nk (µ + τ, Σ) distribution. Example 4.6. Let t → Pt be a differentiable path with score function g and let Tn = Tn (X1 , . . . , Xn ) be statistics such that n  √  1 n Tn − ψ(P ) = √ h(Xi ) + oP (1), n i=1

√  for a function h with P h = 0 and P h2 < ∞. Then the sequence n Tn − ψ(P ) converges in distribution to a normal measure with mean P gh and n√ . variance P h2 under P1/ n  √  Consequently, if ψ is differentiable at P , then n Tn − ψ(P1/√n ) converges in distribution to a normal measure with mean P g(h − ψ˜P ) and variance P h2 . It follows that Tn is a regular estimator sequence if only if h − ψ˜P is orthogonal to the tangent set. In other words if and only if h is an influence function of ψ.

4.2 Gaussian Representations Let H be a subset of a Hilbert space with inner product ·, · and norm ·. For each n ∈ N and h ∈ H, let Pn,h be a probability measure on a measurable space (Xn , An ). Consider the problem of estimating a “parameter” κn (h) given an “observation” Xn with law Pn,h . For ease of notation, let {∆h : h ∈ H} be the “iso-Gaussian process” with zero mean and covariance function E∆h1 ∆h2 = h1 , h2 . The sequence of experiments (Xn , An , Pn,h : h ∈ H) is called asymptotically (shift) normal if log

dPn,h = ∆n,h − 12 h2 , dPn,0

for stochastic processes {∆n,h : h ∈ H} such that 0

∆n,h
∆h

marginally.

h Here
denotes weak convergence under Pn,h . This terminology arises from the theory of limiting experiments due to Le Cam. The sequence of parameters κn (h) is assumed to belong to a Banach space D. We assume that it is asymptotically differentiable in the sense that   ˙ for every h ∈ H, rn κn (h) − κn (0) → κ(h),

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for a continuous, linear map κ: ˙ lin H → D and certain linear maps rn : D → D (“norming operators”). Any maps Tn : Xn → D are considered estimators of the parameter. Example 4.7 (I.i.d. observations). To cover the situation of Lectures 1–3, let Xn = X n , An = An and Pn,h = P1/√n,h for a differentiable path with score h. Furthermore, let κn (h) = ψ(P1/√n,h ). Then differentiability of ψ implies the asymptotic differentiability of κn relative to the norming rate rn = √ n, with derivative κ˙ = ψ˙ P . The asymptotic normality of the experiments n√ : h ∈ P˙ P ) follows from Lemma 1.9, where we may take H = P˙ P , (P1/ n,h contained in the Hilbert space L2 (P ). In all these definitions the measure P is fixed (and considered statistically “known”), and h is an unknown parameter, known to belong to the tangent set. Theorem 4.8 (Gaussian Representation). Suppose that Tn : Xn → D are statistics with values in a Banach space D such that, for every h ∈ H and tight probability measures Lh ,  h √  n Tn − κn (h)
Lh . Assume that the parameters κn are asymptotically differentiable. Then for any orthonormal sequence h1 , . . . , hm in lin H there exists a measurable ˙ is distributed as Lh if the map T : Rm × [0, 1] → D such that T − κ(h) law of T is calculated under the product of the normal measure with mean   h, h1 , . . . , h, hm  and covariance the identity and the uniform measure on [0, 1]. Proof. By an easy calculation we see that the random variable ∆Σai hi −  ai ∆hi has second moment zero. Hence the process h → ∆n,h is linear, in an  almost sure sense. From this we conclude that the sequence ∆n,Σai hi − ai ∆n,hi converges to zero in probability under Pn,0 . Thus the sequence h → ∆n,h is asymptotically linear. Next define variables   Zn,h = rn Tn − κn (h) ,  dPn,h = ∆n,h − 12 h2 , for h = ai hi . Λn (h) = log dPn,0 By assumption, the sequence Zn,0 and each sequence ∆n,h converge in distribution in D and R, respectively. By Prohorov’s theorem, there exists a subsequence of {n} such that 

 0 Zn ,0 , ∆n ,h1 , . . . , ∆n ,hk
(Z, ∆h1 , . . . , ∆hk ),

in D × Rk , where the random vector on the right can be defined on a suitable probability space and has marginal distributions L0 and the standard normal

4.2 Gaussian Representations

375

distribution on Rk , respectively. In view of the asymptotic linearity of the processes h → ∆n,h and the asymptotic differentiability of the sequence of parameters we obtain, for every h ∈ H,

  0 Zn ,h , Λn (h)
Z − κ(h), ˙ ∆h − 12 h2 . Next we can apply Le Cam’s third lemma to see that there exist variables a a Zh , defined on some probability space, such that  Zn ,h
Zh , where
denotes weak convergence under Pn,h for h = ai hi , and Zh is distributed according to


1 2 P(Zh ∈ B) = E1B Z − ˙ i ) e ai ∆hi − 2 a . (4.1) ai κ(h By assumption the weak limit Zh is distributed according to Lh . We are ready to construct an appropriate randomized estimator T . For ease of notation let X0 and U be independent variables with a standard normal distributionon Rk and theuniform distribution, respectively. Suppose  . Then, , . . . , ∆ that T is such that T (X0 , U ), X0 is distributed as (Z, ∆ h h 1 k  if X is normally distributed with mean vector µ(h) = h, h1 , . . . , h, hm  and covariance the identity and independent of U , we have 1 T     T Ph T (X, U ) ∈ B) = Eh 1B T (X, U )) = E0 1B T (X, U ) eµ(h) X− 2 X X

= Lh (B),   because (X, U ) is distributed as (X0 , U ) under h = 0 and hence T (X, U ), X is distributed as (Z, ∆h1 , . . . , ∆hk . To conclude the proof it suffices to construct T as in the preceding para-   graph. Because the second marginal distributions of the vectors T (X , U ), X0 0  and (Z, ∆h1 , . . . , ∆hk are identical, it suffices to construct T such that the conditional distributions of the first marginal given the second marginal are identical. This is the case if for each x0 the variable T (x0 , U ) is distributed according the conditional law of Z given (∆h1 , . . . , ∆hk ) = x0 . This is the problem of generating a variable with an arbitrary distribution on a Polish space from a uniform variable. It is well-known that this possible. One construction is to map the Polish space bimeasurably onto the real line, and next use the quantile transformation to construct the induced law.

The preceding theorem is restricted to finite-dimensional models. As we remarked before an extension to infinite-dimensional Hilbert spaces H is possible, but maybe not very useful, because it is hard to analyse infinitedimensional Gaussian experiments directly, without finite-dimensional approximation. For completeness we include an infinite-dimensional version. Theorem 4.9. Suppose that Tn : Xn → D are statistics with values in a Banach space D such that, for every h ∈ H and a tight probability measure Lh ,

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4 Lecture: Gaussian Approximations

 h √  n Tn − κn (h)
Lh . Assume that the parameters κn are asymptotically differentiable. Then for any orthonormal sequence h1 , h2 , . . . , in lin H there exists a measurable ˙ is distributed as Lh if the map T : R∞ × [0, 1] → D such that T − κ(h) law of T is calculated under the product of the normal measure with mean   h, h1 , h, h2 , . . . and covariance the identity and the uniform measure on [0, 1]. The infinite-dimensional normal measure in the theorem is simply the distribution of a sequence Z = (Z1 , Z2 , . . .) of independent normal variables Zi with means h, hi  and unit variances. Actually, the Gaussian experiment could be represented in many different forms, the present one probably being the simplest one. For instance, the theorem is also true if Z is replaced by f (Z) for an arbitrary bimeasurable map f from R∞ onto a measurable space (e.g. the unit interval). The preceding theorems show that estimator sequences in the sequence of experiments (Pn,h : h ∈ H) are asymptotically matched by an estimator in a Gaussian experiment. The next step is to analyse the Gaussian experiment. In our abstract set-up the “optimal” measure can be defined in terms of the adjoint κ˙ ∗ : D∗ → lin H of the asymptotic derivative of the parameters κn , which maps the dual space of D into the closed linear span of H. This is ˙ determined by the equation κ˙ ∗ b∗ , h = b∗ κ(h). The optimal Gaussian measure can be uniquely determined by its marginal distributions: its induced laws under continuous, linear maps d∗ : D → R. It can be represented as the distribution of  a Borel measurable random element  G in D such that d∗ G is N 0, κ˙ ∗ d∗ 2 distributed, for any element d∗ from the dual space D∗ . In the case that the Banach space D is infinite-dimensional such a measure does not necessarily exist, but the the theorem below shows that it does exist when we need it: if there exist good estimator sequences. Example 4.10. Consider the case of i.i.d. observations as considered in Lecture 2, with κ˙ = ψ˙ P , H = ψ˙ P equipped with the L2 (P )-norm, and ψ taking values in D = Rk . Then κ is representable as a vector-valued inner product ψ˙ P (h) = P hψ˜P , D∗ = Rk , and its adjoint is the map κ˙ ∗ a = aT ψ˜P , because ˙ = aT P hψ˜P = P (aT ψ˜P )h. P (κ˙ ∗ a)h = a, κh It follows that the optimal limit measure is the distribution of a vector G such that aT G is normally distributed with mean 0 and variance aT ψ˜P 2 = P (aT ψ˜P )2 . This agrees with the optimal normal measure found in Lecture 2. In our present set-up we call a sequence of estimators Tn regular with respect to the norming operators rn if   h rn Tn − κn (h)
L, for every h ∈ H, for a fixed, tight, Borel probability measure L on D.

4.2 Gaussian Representations

377

Theorem 4.11 (Convolution). Assume that the parameters κn are asymptotically differentiable. (i)

If there exists a sequence of regular estimators for κn , then there exists a tight, Borel measurable variable G in D such that   for every d∗ ∈ D∗ . d∗ G ∼ N 0, κ˙ ∗ d∗ 2 ,

(ii) The limit law L of every regular sequence of estimators can be represented as the distribution of a sum G + W of independent, tight, Borel measurable variables in D with G as distributed in (i). Proof. (a) Assume that H is a finite-dimensional, linear space and let h1 , . . . , hm be an orthonormal base. Then the assumptions of Theorem 4.8 are satisfied and we obtain that L is the distribution of T − κ(h) ˙ under every h. As shown in the proof of this theorem (see (4.1)), this means that, for every a ∈ Rk ,


1 2 L(B) = E1B Z − ˙ i ) e ai ∆hi − 2 a . ai κ(h We average this equation over a with respect to a Nk (0, λ−1 I) weight function. Straightforward calculations yield        ai κ(h ˙ i) ˙ i) ∆hi κ(h − cλ (∆) dNk 0, I (a), L(B) = E1B Z − 1/2 1+λ (1 + λ)    where cλ (∆) = (1 + λ−1 )k/2 exp 12 (1 + λ)−1 ∆2hi . Conclude that L can be written as the law of the  sum Gλ + Wλ of independent random elements ˙ i )/(1 + λ)1/2 for a Nk (0, I)-distributed Gλ and Wλ , where Gλ = − Ai κ(h vector (A1 , . . . , Ak ) and Wλ is distributed according to    ˙ i) ∆hi κ(h cλ (∆). P(Wλ ∈ B) = E1B Z − 1+λ  Ai κ(h ˙ i ). The variable d∗ G =  As ∗λ ↓ 0, we have Gλ
G = ˙ i ) is normally distributed with zero mean and variance Ai d κ(h  2 d∗ κ(h ˙ i ) = κ˙ ∗ d∗ 2 . Ed∗ G2λ = By the converse part of Prohorov’s theorem, the variables {Gλ : 0 < λ < 1} are uniformly tight. Combined with the tightness of L it follows that there exists, for every ε > 0, a compact K such that  1 − ε < L(K) = P(Wλ + g) dP Gλ (g), and P(Gλ ∈ K) > 1 − ε. This implies that for every λ there exists gλ ∈ K such that P(Wλ + gλ ∈ K) > 1 − 2ε and hence P(Wλ ∈ K − K) > 1 − 2ε). We conclude that set of the variables {Wλ : 0 < λ < 1} is uniformly tight.

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If Wλm
W for a sequence λm ↓ 0, then (Gλm , Wλm )
(G, W ), where G and W are independent and G + W is distributed according to L. This concludes the proof of the theorem for finite-dimensional H. (b) Let H be arbitrary. For any finite orthonormal set h1 , . . . , hk , the previous argument yields tight independent processes Gk and Wk such that Gk + Wk is distributed according to L and Gk is zero-mean Gaussian with Ed∗ G2k =

k

κ˙ ∗ d∗ , hi 2 .

i=1

The set of all variables Gk and Wk so obtained is uniformly tight. Indeed, by tightness of
L, there exists for any given ε > 0 a compact set K such that L(K) = P(Gk ∈ K − x) dP Wk (x) > 1 − ε. Thus there exists x0 with P(G  k ∈ K − x0 ) > 1 − ε. By symmetry, P(Gk ∈ x0 − K) > 1 − ε, whence P Gk ∈ 12 (K − K) > 1 − 2ε. Next, the uniform tightness of L and the collection Gk imply the uniform tightness of the collection Wk . Direct the finite-dimensional subspaces of H by inclusion, and construct variables (Gk , Wk ) for every subspace. Every weak limit point (G, W ) of the

net of laws (Gk , Wk ) satisfies the requirements of the theorem. In the following minimax theorem we show that the maximum risk   sup Eh∗  rn (Tn − κn (h)) h

of an estimator sequence can never asymptotically fall below E(G). A little (asymptotic) measurability is the only requirement on Tn , but measurability can be restrictive, so we shall be careful about it. Let D be a given subspace of D∗ that separates points of D, and let τ (D ) be the weak topology induced on D by the maps b : D → R when b ranges over D . Definition 4.12. A map : D → R is called τ (D )-subconvex if for every  c > 0 the set y: (y) ≤ c is τ (D )-closed, convex, and symmetric. Theorem 4.13 (Minimax theorem). Assume that the parameters κn are asymptotically differentiable. Suppose a tight, Borel measurable Gaussian element G as in (i) of the statement of the convolution theorem exists. Then for every estimator sequence Tn such that d Tn : Xn → R is measurable for every d ∈ D and every τ (D )-subconvex function : D → R,   sup lim inf sup Eh∗  rn Tn − κn (h) ≥ E(G). I⊂H n→∞ h∈I

Here the first supremum is taken over all finite subsets I of H. Proof. In a general sense the proof is based on an analysis of the minimax risk in the Gaussian representation provided by Theorem 4.8. The main work is to

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379

force our estimator sequence to have limit laws, so that the theorem becomes applicable. This is achieved by compactification of the range space of Tn , so that limit laws exist at least along subsequences and with limits concentrating on the compactification, by Prohorov’s theorem. Because it will be necessary to extend the loss function to the compactification, the compactification must be chosen dependent on the loss function. Therefore the proof proceeds in several steps, building more complicated loss functions from simple ones. (a)  Assume first  that the loss function can be written in the special form r (y) = i=1 1Kic di,1 y, . . . , di,pi y for compact, convex, symmetric subsets Ki ⊂ Rpi and arbitrary elements di,j of D . Fix an arbitrary orthonormal set h1 , . . . , hk in H, and set      i = di,1 , . . . , di,pi ◦ rn Tn − κn ( ai hi ) , 1 ≤ i ≤ r. Zn,a Considered as maps into the one-point compactification of Rpi , the sequences i are certainly asymptotically tight. The sequences are asymptotically Zn,a measurable by assumption.  Direct  the finite subsets of H by inclusion. There exists a subnet nI : I ⊂ H, finite such that the left side of the statement of the theorem equals   minimax risk = lim sup Eh∗  rn Tn − κn (h) . I

h∈I

By the same arguments as in the proof of the convolution theorem there a Zai in the one-point is a further subnet {n } ⊂ {nI } such that Zni  ,a
k compactifications, for every a ∈ R and every i. Here the limiting processes satisfy, for each i,  (4.2) L(Zai ) dNk (0, λ−1 I) ∼ Giλ + Wλi , for independent elements Giλ and Wλi such that Giλ

=



di,1 , . . . , di,pi



◦ Gλ =



di,1 , . . . , di,pi



 ◦

˙ i) Ai κ(h , (1 + λ)1/2

for a Nk (0, I)-distributed vector (A1 , . . . , Ak ). By the portmanteau theorem, minimax risk ≥ lim inf n

r

Pa∗ (Zni  ,a ∈ / Ki ) ≥

i=1

r

P(Zai ∈ / Ki ).

i=1

Since this is true for every a, the left side is also bounded below by the average of the right side, in particular the average under the Nk (0, λ−1 I)-distribution. In view of (7.4) we find that minimax risk ≥

r i=1

P(Giλ + Wλi ∈ / Ki ).

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The right side becomes smaller if we replace the variables Wλi by 0. This follows by Anderson’s lemma, according to which, given a mean zero Gaussian vector and a convex, symmetric set K, the probability P(G + a ∈ K) is maximized over a at a = 0, i.e. centering the Gaussian variable G + a at zero. Thus the right side of the preceding display is bounded below by r

P(Giλ ∈ / Ki ) = E(Gλ ).

i=1

We finish the proof for this special form of loss function by letting λ ↓ 0 followed by taking the limit along finite-dimensional subspaces of H. (b) The theorem is “closed” under taking monotone limits on : if the theorem holds for every function r and 0 ≤ r ≤  with r ↑  almost surely under the law of G, then the theorem holds for . To see this, note that the minimax risk decreases by replacing  by r . Thus it is bounded below by Er (G) for every r, which increases to E(G) as r → ∞. (c) An arbitrary subconvex  can be approximated from below by a sequence of functions r of the type as in (a). To see this, note first that 2r

−r

0≤2

2   1 y: (y) > i2−r ↑ (y),

for every y.

i=1

  Each of the sets y: (y) > i/r is convex, τ (D )-closed, and symmetric. Thus, it suffices to approximate functions  of the type 1C c for a convex, τ (D )-closed, and symmetric set C. By the Hahn–Banach theorem, any such set C can be written    y: |b y| ≤ cb . C= b ∈D

Thus the complement of C intersected with the support S of the limit variable   G is the union of the sets y ∈ S: |b y| > cb . These sets are relatively open in S and S is separable. Since a separable set is Lindel¨of, the possibly uncountable union can be replaced by a countable subunion. exists  Thus there   a sequence di in D and numbers ci such that C c ∩S = ∪∞ y ∈ S: |d y| > ci . i=1 i This implies that   1C c ∩S = sup 1Krc d1 y, . . . , dr y , r

 

for the subsets of R defined by Kr = ∩ri=1 x ∈ Rr : |xi | ≤ ci . r

Example 4.14. For D = D∗ , the τ (D )-topology is the weak topology. Because convex subsets in a Banach space are weakly closed if and only if they are closed for the norm, a function which is subconvex relative to the norm is automatically τ (D∗ )-subconvex. The theorem is applicable to the combination of such loss functions and estimator sequences Tn that are weakly measurable: d∗ Tn should be a measurable map in R for every d∗ ∈ D∗ .

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381

This will typically be the case if the Banach space is separable, when estimators will usually be required to be Borel measurable. Example 4.15 (Skorohod space). The Skorohod space D[a, b], for a given interval [a, b] ⊂ R, is a Banach space if equipped with the uniform norm. The dual space consists of maps of the form d∗ (z) =

 z(u) dµ(u) +

∞

  αi z(ui ) − z(ui −) ,

i=1

for a finite signed measure µ on [a, b], an arbitrary sequence ui in (a, b], and  |αi | < ∞. (This is an extension of the representation a sequence αi with theorem for the dual space of the space of continuous functions on a compact due to Riesz, obtained in [36, pp. 81–85]) Each such d∗ is the pointwise limit of a sequence of linear combinations of coordinate projections. Thus, the σ-field generated by the dual space equals the σ-field generated by the coordinate projections. It follows that an estimator sequence is D[a, b]∗ -measurable if and only if it is a stochastic process. Since “τ (D[a, b]∗ )-subconvex” is identical to “subconvex with respect to the norm”, the minimax theorem is valid for any sequence of stochastic processes Tn and subconvex loss function . Examples of subconvex loss functions include   z → 0 z∞ ,  z → |z|p (t) dµ(t), for a nondecreasing, left-continuous function 0 : R → R, a finite Borel measure µ, and p ≥ 1. Example 4.16 (Bounded functions). On the space ∞ (F), functions of the type  z    z → 0   , q F for a nondecreasing, left-continuous function 0 : R → R and an arbitrary map q: F → R are subconvex with respect to the linear space spanned by the coordinate projections z → z(f ). Indeed, for any c there exists d such that         z       z  z: 0   ≤ c = z:   ≤ d = z: z(f ) ≤ d q(f ) . q F q F f ∈F

Thus, the minimax theorem is valid for any estimator sequence Tn that is coordinatewise measurable and any loss function of this type. For general loss functions that are subconvex with respect to the norm, the preceding minimax theorem applies only under strong measurability conditions on the estimator sequences. It is of interest that these measurability

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conditions are satisfied by sequences Tn such that  Tn (f ) is measurable for every f and such that the sequence rn Tn −κn (0) is asymptotically tight under Pn,0 . Indeed, such sequences are asymptotically τ (∞ (F)∗ )-measurable. It follows that, given any subconvex loss function, the minimax theorem may be used to designate optimal estimator sequences among the asymptotically tight sequences. Finally, consider the testing problem. The Gaussian representation theorem given previously was meant to be applied to the estimation problem, but we can easily transform it into a theorem on tests by taking κn (h) ≡ 0. Theorem 4.17 (Gaussian Representation). Let φn : Xn → [0, 1] be arbitrary statistics such that, for every h ∈ H and some function π: H → R, Pn,h φn → π(h). Then for any orthonormal sequence h1 , . . . , hm in lin H there exists a measurable map φ: Rm → [0, 1] such that  Ph φ = π(h) for every  h ∈ H, where Ph is the normal measure with mean h, h1 , . . . , h, hm  and covariance the identity. Proof. Because the unit interval is compact we can extract a subsequence of φn that converges in distribution under Pn,h to a limit law Lh . By contiguity arguments, using Le Cam’s third lemma as in the proofs of the preceding theorems, we can even find a subsequence that works for all h in the linear span of h1 , . . . , hm . We next apply Theorem 4.8 to the corresponding subsequence of κn ≡ 0 and Tn = n−1/2 φn to find that there exist a randomized estimator T with values in [0, 1] that has law Lh under the product of Ph and the uniform measure. Then φ(x) = ET (x, U ) has the desired properties.

The message of the theorem is that every limiting power function is necessary the power function of a test in the limiting Gaussian experiment. The assumption that there exists a limiting power function is very weak, because by the compactness of the unit interval we can always construct subsequences along which a limit exists. An analysis of tests in the Gaussian experiment yields concrete bounds on, for instance, the power of level α tests. Compare Theorem 2.11 in Lecture 2. Open problem. The theorems as presented in this lecture apply to many time series models. However, the semiparametric theory for such models, e.g. discretely observed diffusion processes, appears to be largely undeveloped. Notes This lecture is based on [36], [38] and Chapter 3.11 of [41]. It is strongly motivated by ideas of Le Cam, in particular from his papers [15], [16] and [17], and earlier results by [20] and [22] and [23].

5. Lecture: Empirical Processes and Consistency of Z-Estimators

In this lecture and the next lecture we discuss empirical processes. Our main focus is the application of empirical processes to the derivation of asymptotic properties of estimators in semiparametric models. In this first lecture we discuss entropy numbers, Glivenko–Cantelli classes and their application to proving consistency of M - and Z-estimators.

5.1 Empirical Measures and Entropy Numbers Given i.i.d. random variables X1 , . . . , Xn with law P on a measurable space (X , A) and a measurable function f : X → R we let 1 f (Xi ), n i=1  P f = f dP, n

Pn f =

 √ 1  f (Xi ) − P f = n(Pn − P )f, Gn f = √ n i=1 1/r  f P,r = P |f |r . n

Given a class F of measurable functions f : X → R we view Pn as a map f → Pn f on F. Of course, we can also think of Pn as the discrete uniform random measure on the points X1 , . . . , Xn . We denote by F a measurable  envelope function of the class F: a function F : X → R such that f (x) ≤ F (x) for every x ∈ X and f ∈ F. For a function z: F → R the norm zF is the  supremum norm: zF = supf ∈F z(f ). The law of large numbers asserts that Pn f → P f almost surely if P f exists, and the central limit theorem asserts that Gn f is asymptotically normal if P f 2 < ∞. An important aim in empirical process theory is to make these statements uniform in f ranging over a class F, in an appropriate sense. We shall also be concerned with the behaviour of Pn fˆn and Gn fˆn for fˆn a “random function”, which is related to uniformity. Uniformity over a class of functions depends on the size of a class. An appropriate measure of size are entropy numbers, which come in two types:

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with or without bracketing. Given two measurable functions l, u: X → R, the bracket [l, u] is the collection of all functions f : X → R such that l ≤ f ≤ u. Let  ·  be a norm on a collection of functions. An ε-bracket is a bracket [l, u] such that u − l < ε. Here it is required that both l and u are of finite norm.   Definition 5.1. The bracketing number N[] ε, F, · is the smallest number of ε-brackets needed to cover F.   Definition 5.2. The covering number N ε, F,  ·  is the smallest number of balls of radius ε needed to cover F. The logarithms of bracketing or covering numbers are called entropies. An ε-bracket [l, u] is contained in a ball of radius ε/2 around  the midpoint  1 ε, F,  ·  and (l + u) of the bracket. It follows that N ε/2, F,  ·  ≤ N [] 2 hence bracketing numbers are bigger than covering numbers (the factor 2 is of no importance in the following). On the other hand, brackets give pointwise control over functions, whereas for many norms knowing that some function is in a ball, even a very small one, still leaves irregular behaviour on a set of small measure open. Such small sets are important when the function is applied to random variables Xi . This observation explains that typically conditions using covering numbers use many different norms simultaneously, whereas conditions using bracketing numbers use the “true” law P only. The best results using covering numbers are in terms of random covering numbers. For simplicity, we state the results in terms of the bigger uniform covering numbers. Definition 5.3. The Lr -uniform covering numbers relative to  the envelope function F are the numbers supQ N εF Q,r , F,  · Q,r , where the supremum is taken over all discrete probability measures Q on (X, A) with F Q,r > 0.   A class F is, by definition, totally bounded if and only if N ε, F, · < ∞ for every ε > 0. (Then its completion is compact.) This will be necessary for the desired uniform law of large numbers or central limit theorem to hold, but it is by far not enough. A more precise measure of the size of a class F is the rate at which the covering or bracketing numbers increase as ε ↓ 0.

5.2 Glivenko–Cantelli Classes The Glivenko–Cantelli theorem is the uniform version of the law of large numbers. The classical Glivenko–Cantelli theorem concerns the uniformity in the convergence of the empirical cumulative distribution function of realvalued random variables. The abstract version is named after this. Definition 5.4. A collection F of measurable functions f : X → R is P Glivenko–Cantelli if Pn − P F → 0 almost surely.

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385

We note that the random distance Pn − P F need not be measurable. By “almost sure” convergence Zn → Z of a sequence of possibly unmeasurable maps with values in a metric space, we shall understand that there exist measurable maps ∆n on the underlying probability space such that d(Zn , Z) ≤ ∆n for each n and ∆n → 0 almost surely. Two basic theorems on Glivenko–Cantelli classes are as follows.   Theorem 5.5. If N[] ε, F,  · P,1 < ∞ for every ε > 0, then F is P Glivenko–Cantelli.   Theorem 5.6. If supQ N εF Q,1 , F,  · Q,1 < ∞ for every ε > 0, P F < ∞ and F is suitably measurable, then F is P -Glivenko–Cantelli. The condition that F be “suitably measurable” will recur, but what is suitable will depend on the situation. In the present case it may be taken to mean that the suprema n 1    ei f (Xi )  n i=1 F are measurable, for every fixed vector (e1 , . . . , en ) ∈ {−1, 1}n , and every n ∈ N. A simple sufficient condition for this is that the supremum be equal to the same supremum but then computed over a countable class F, e.g. a subclass G ⊂ F. The suitable measurability is necessary because the proof of the theorem is based on a symmetrization and conditioning device, requiring an application of Fubini’s theorem. The second, uniform entropy theorem is much harder to prove than the bracketing Glivenko–Cantelli theorem, which can be modelled after the proof of the classical Glivenko–Cantelli theorem. The condition of the first theorem implies that P F < ∞: if we cover F with finitely many brackets, of for instance size 1, and next take the supremum of the absolute values of all upper and lower bracketing functions, we obtain an integrable envelope. Thus the difference between the two theorems resides solely in the use of bracketing or covering numbers. The stronger bracketing numbers may be replaced by the weaker covering numbers, but only at the cost of using uniform covering numbers. Upper bounds on the covering or bracketing numbers of many classes of functions are known from the classical references on these subjects (1950/60s), from more recent work in approximation theory, and from the combinatorial theory employed by Vapnik and Chervonenkis. Statistical problems, in particular in semiparametric modelling, generate many new classes of functions, sometimes of a complicated nature, for which such estimates are not known. Then we must either derive new estimates or can use stability theorems that allow the construction of new Glivenko– Cantelli classes from known Glivenko–Cantelli classes. The following theorem is in this spirit and can save much work.

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For ease of terminology we call a collection of measurable functions f : X → Rk Glivenko–Cantelli if each of the k collections of coordinate functions is Glivenko–Cantelli. Theorem 5.7. If F is a Glivenko–Cantelli class of functions f : X → Rk with integrable envelope and φ: Rk → R is continuous, then the class of functions φ ◦ f : X → R is Glivenko–Cantelli provided that it has an integrable envelope.

5.3 Consistency of M- and Z-estimators Glivenko–Cantelli classes are useful to carry out proofs that M - or Zestimators are consistent. These are estimators defined to be a point of maximum or a zero of a given stochastic process. To remain within the theme of empirical processes, we restrict ourselves to criterion functions that are averages over the observations. For every θ in a metric space Θ let mθ : X → R be a measurable function. An M -estimator θˆn is a point of maximum of the map θ → Pn mθ . The aim is to show that this converges in probability to a point of maximum θ0 of the map θ → P mθ . The following theorem states a stronger result. Theorem 5.8. Suppose that the class of functions {mθ : θ ∈ Θ} is P Glivenko–Cantelli and that there exists a point θ0 ∈ Θ such that inf θ:d(θ,θ0 )>δ P mθ < P mθ0 for every δ > 0. Then Pn mθˆn ≥ Pn mθ0 implies that d(θˆn , θ0 ) → 0 almost surely. Proof. By the property of θˆn , we have Pn mθˆn ≥ Pn mθ0 = P mθ0 − o(1), almost surely. Hence P mθ0 − P mθˆn ≤ Pn mθˆn − P mθˆn + o(1) ≤ sup |Pn mθ − P mθ | + o(1) → 0, θ

almost surely. By assumption there exists for every δ > 0 a number η > 0 such that P mθ < P mθ0 − η for every θ with  d(θ, θ0 ) > δ. Thus,  the event  d(θˆn , θ0 ) ≥ δ is contained in the event P mθˆn < P mθ0 − η . The latter sequence of events decreases to a zero event, in view of the preceding display.

This theorem is good enough for most purposes, but can be improved in two important ways: – As is clear from the proof, the double-sided convergence given by the Glivenko–Cantelli property is used only to ensure a one-sided convergence, corresponding to the fact that we maximize a criterion function. However, we like the simple Glivenko–Cantelli condition over a more complicated one-sided condition. The tricks that we present below often blur the difference.

5.3 Consistency of M- and Z-estimators

387

– If θ is far from θ0 , then usually P mθ will be far from P mθ0 . The closeness of the random criterion Pmθ to the limit P mθ need therefore not be uniform in θ as it is required by the Glivenko–Cantelli property. We shall make use of this in Lecture 8 when discussing rates of convergence. (It appears that any type of relaxation of the Glivenko–Cantelli condition to make this point precise, automatically results in a stronger statement concerning a rate of convergence.) Thus we shall not formulate any refinements here. Our interest will go in another direction: application to semiparametric estimation problems. Before discussing a concrete example, it is instructive to compare the present theorem to the one obtained by Wald in the 1940s. (Wald had maximum likelihood estimators in mind, but his proof applies equallly well to general M -estimators.) Wald’s main conditions were compactness of the parameter set (or the possibility of compactification) and local domination. Taking the preceding remarks into account the present theorem contains Wald’s theorem, in view of the following lemma. Lemma 5.9. Let Θ be a compact metric space, let the map θ → mθ (x) be continuous for every x ∈ X and suppose that every θ has a neighbourhood B such that supθ∈B |mθ | is dominated by an integrable function. Then the class {mθ : θ ∈ Θ} is Glivenko–Cantelli and inf θ:d(θ,θ0 )>δ P mθ < P mθ0 for every δ > 0 if and only if θ → P mθ possesses a unique global maximum at θ0 . Proof. The compactness of Θ and the local domination of the functions mθ imply that the class {mθ : θ ∈ Θ} possesses an integrable envelope function. The dominated convergence and the assumed continuity of the maps θ → mθ (x) imply that the map θ → P mθ is continuous. Thus it attains its maximum on the compact set {θ ∈ Θ: d(θ, θ0 ) ≥ δ} for every given δ > 0, and this is smaller than its value at θ0 , by the assumption that θ0 is a unique maximum. To complete the proof we show that the L1 (P )-bracketing numbers of the class {mθ : θ ∈ Θ} are finite. If Bm is a decreasing sequence of neighbourhoods of a fixed θ such that ∩m Bm = {θ} and um and lm are defined as the supremum and infimum of the functions mθ with θ ∈ Bm , then um − lm → mθ − mθ = 0 as m → ∞, by the continuity of the functions θ → mθ . By the dominated convergence theorem P (um −lm ) → 0. We conclude that for every ε > 0 and θ ∈ Θ there exists a neighbourhood B such that P (uB − lB ) < ε, for uB and lB the supremum and infimum of the functions mθ with θ ∈ B. The collection of neighbourhoods B obtained this way by varying θ over Θ has a finite subcollection that covers Θ, by the compactness of Θ. The corresponding brackets [lB , uB ] cover the class {mθ : ∈ Θ}.

The preceding theorem reduces the consistency proof of an M -estimator to verification of the good behaviour of the limit criterion function θ → P mθ and a Glivenko–Cantelli property. The same methods apply to Z-estimators.

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For every θ in a set Θ ⊂ Rk let ψθ : X → Rk be a measurable, vectorvalued function. A Z-estimator θˆn is a zero of the map θ → Pn ψθ . The aim is to show that this converges in probability to a zero θ0 of the map θ → P ψθ . Theorem 5.10. Suppose that the class of functions {ψθ : θ ∈ Θ} is P Glivenko–Cantelli and that there exists a point θ0 ∈ Θ such that inf θ:d(θ,θ0 )>δ P ψθ  > 0 = P ψθ0  for every δ > 0. Then Pn ψθˆn = 0 implies that d(θˆn , θ0 ) → 0 almost surely. Proof. By the Glivenko–Cantelli property P ψθˆ = Pn ψθˆ + o(1) = o(1), ˆ Thus it is impossible that almost surely as n → ∞, by the property of θ. ˆ d(θ, θ0 ) > δ infinitely often, for any δ > 0.

Not with standing beautiful and simple results as the preceding theorems, it remains an unfortunate fact that consistency proofs are not easily forced into a single mould. Because consistency concerns the behaviour of estimators on the global model, a differential analysis, such as possible for normality proofs, is impossible. (Unless one is satisfied with statements as: there exists some sequence of local maxima that converges to a true value, without worrying about the selection of such a sequence or the behaviour of an arbitrary sequence of maxima. We are not.) Proving consistency remains somewhat of an art, and is sometimes the hardest part of the analysis of a maximum likelihood estimator. This is true in particular for semiparametric maximum likelihood estimators, because semiparametric likelihoods may be ill-behaved. In the following three sections we discuss some useful tricks, each time illustrated by an example of interest. 5.3.1 Trick 1: Errors-in-variables  
Consider the errors-in-variables models pθ,η (x, y) = φ(x − z)φ y − fθ (z) dη(z), where φ is the standard normal density. The regression function fθ is assumed known up to a parameter θ ∈ Θ ⊂ Rk . We wish to prove ˆ that n the maximum likelihood estimator (θ, ηˆ) defined as the maximizer of i=1 pθ,η (Xi , Yi ) over all θ ∈ Θ and probability distributions η on some interval Z ⊂ R is consistent. To simplify we assume that Θ and Z are compact. In the case that the natural parameter space for z is the real line, we could achieve this by extending the model to all probability distributions on the extended real line  R, defining φ(x − z)φ y − fθ (z) to be zero for z = ±∞. Furthermore, we assume that (θ, z) → fθ (z) is continuous on Θ × Z. The set of all probability measures on Z is compact under the weak topology. Furthermore, the map (θ, η) → pθ,η (x, y) is continuous for every (x, y). To analyse the maximum likelihood estimator we might apply the preceding theorem with the functions mθ,η = log pθ,η . These would form a Glivenko– Cantelli class by the preceding lemma, except for the fact that we need to

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389

find an integrable envelope function. These functions are bounded above, but their unboundedness from below could prevent this from being true. Because we are interested in a point of maximum, unboundness from below should not cause problems. We could see this by improving the preceding theorem, along the lines of the remarks following its proof. A simpler approach is to apply the theorem not with the functions log pθ,η , but with the functions

p + p θ,η θ0 ,η0 . mθ,η = log 2 ˆ ηˆ) maximizes Pn mθ,η It is not true that the maximum likelihood estimator (θ, for this choice of mθ,η , but it is true that Pn mθ,ˆ ˆ η = Pn log

p ˆ + pθ ,η 0 0 θ,ˆ η 2



≥ Pn 21 log pθ,ˆ ˆ η + log pθ0 ,η0 ≥ Pn log pθ0 ,η0 .

In the first inequality we use the concavity of the logarithm, and in the second ˆ ηˆ). Thus Pm ˆ ≥ Pmθ ,η and this is good enough for the definition of (θ, 0 0 θ,ˆ η the application of the theorem, because we also have that P mθ,η < P mθ0 ,η0 unless the densities 12 (pθ,η +pθ0 ,η0 ) and pθ0 ,η0 define the same measure. Equivalently, unless pθ,η and pθ0 ,η0 define the same probability measure. The last requirement concerns the identifiability of the parameter (θ0 , η0 ). This depends on the nature of the functions fθ and is a nontrivial matter. For the case of linear functions fθ it was settled in the 1970s. The functions pθ,η and hence the functions mθ,η are uniformly bounded above. Furthermore, the functions mθ,η are bounded below by the function log pθ0 ,η0 − log 2. Hence the class of functions mθ,η has a Pθ0 ,η0 -integrable envelope if Pθ0 ,η0 log pθ0 ,η0 > −∞. By Jensen’s inequality     Pθ0 ,η0 (− log pθ0 ,η0 ) ≤ Pθ0 ,η0 (− log) φ(x − z)φ y − fθ0 (z) dη0 (z)    Eθ0 ,η0 X 2 + Y 2 + Z 2 + fθ0 (Z)2 . The right side is finite under reasonable assumptions on η0 . 5.3.2 Trick 2: Cox model In many semiparametric models “likelihoods” are defined through point masses. A Wald-type proof of consistency is then ruled out by the lack of continuity of the likelihood relative to a useful topology. A proof of consistency may then proceed by an intermediate step using “likelihood equations”, but still relying on the Glivenko–Cantelli theorem at several points. We illustrate this for the Cox model, as described in Lecture 3, Example 3.10. Other models have been treated by the same method, albeit that the exact arguments usually are more complicated. We make the same assumptions as

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in Lecture 3. In particular, C is smaller than some τ with probability one and satisfies P(C = τ ) > 0 and P(T > τ ) > 0. The density of an observation in the Cox model takes the form

eθz λ(y)e−e

θz

Λ(y)



1−δ  δ −eθz Λ(y) e 1 − FC|Z (y−| z) fC|Z (y| z) pZ (z).

To define a maximum likelihood estimator for (θ, Λ), we of course drop the terms involving the distribution of (C, Z), which is assumed not to depend on the parameter of interest. Unfortunately, the supremum of n 

eθZi λ(Yi )e−e

θZi

Λ(Yi )

∆ i

e−e

θZi

Λ(Yi )

1−∆i

i=1

over all parameters θ and hazard functions λ is infinite. We can approximate this supremum by choosing hazard functions that have very high, but very thin peaks around the values Yi with ∆i = 1. By making the peaks sufficiently thin we can ensure that the values Λ(Yi ) are arbitrarily close to zeroand hence n the value of the preceding display will be determined by the factor i=1 λ(Yi ). Thus we cannot define a maximum likelihood estimator in this way. The way out is to define the likelihood instead by n 

eθZi Λ{Yi }e−e

θZi

Λ(Yi )

∆i

e−e

θZi

Λ(Yi )

1−∆i

i=1

Next we maximize over all θ ∈ Θ and nondecreasing, cadlag functions Λ: [0, ∞) → R with Λ(0) = 0. (This is a bit bigger than the set of cumulative hazard functions, defined as finite measures of the type dΛ = dF/(1 − F −) for cumulative distributions F , which are restricted to having jumps of size less than 1, but asymptotically this will not make a difference.) Maximizing relative to Λ entails maximizing the jumps Λ{Yi } at points Yi with ∆i = 1, meanwhile minimizing the cumulative masses Λ(Yi ) at Yi such that ∆i = 0. The best choice is among the discrete distributions Λ that jump at the points Yi with ∆i = 1 only. This observation reduces the maximization problem to a finite-dimensional one (finding the jump sizes), and the compactness of the unit simplex implies that a solution exists, also jointly in θ and Λ. What we have called “likelihood” does not have the continuity property we would require for a Wald type proof. Also the parameter space for Λ is not a-priori compact. We get around this problem by using likelihood equations. For a bounded function h we can define by dΛˆt = (1 + th) dΛˆ a perturbation ˆ defined for at least every t in a neighbourhood of 0. The likelihood of Λ, ˆ Λˆt ) viewed as a function of t must be maximal at t = 0. evaluated at (θ, Differentiating at t = 0 we obtain the stationary equation Pn Bθˆn ,Λˆn h = 0,

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where Bθ,Λ is (the version of) the score operator given in Example 3.10. We can rewrite this equation as   ˆ ˆ Pn δh(y) = Pn eθn z h dΛˆn = Pn eθn z h(s)1s≤y dΛˆn (s). [0,y]

In this notation Pn is the empirical measure of the observations Xi = (Yi , ∆, Zi ), and we write Pn f (x) instead of Pn f for clarity (we hope). Inˆ n,0 ), we find verting the preceding display (i.e. replacing h by h/M ˆ n = Pn δh(y) , Λh ˆ n,0 (y) M

ˆ ˆ n,0 (s) = Pn eθz 1s≤y . M

If we knew that θˆn were consistent, then we could use this representation directly to prove the consistency of Λˆn . The Cox model, as usual, is much simpler here than other models. In other situations we find a recursive expression for Λˆn with both Λˆn and θˆn appearing on the right side, but the argument may proceed in the same way. The Wald argument is based on comparing the value of the likelihood at the maximum likelihood estimator and at the true value of the parameter. In the present case this causes a problem, because the likelihood at the maximum likelihood estimator, a random discrete distribution, and at the true parameter are different in character. This is solved by comparing the likelihood at the maximum likelihood estimator and at the random parameter (θ0 , Λ˜n ) for Λ˜n defined by δh(y) Λ˜n h = Pn , M0 (y)

M0 (s) = P0 eθ0 z 1s≤y .

The function Λ˜n is similar in structure to Λˆn , but is also similar to Λ0 : applying the same algebra as previously to the equation P0 B0 h = 0 we see that δh(y) . Λ 0 h = P0 M0 (y) Under our assumptions M0 (s) ≥ M0 (τ ) is bounded away from zero. Therefore, the functions (y, δ) → δh(y)/M0 (y) form a Glivenko–Cantelli class if h ranges over a Glivenko–Cantelli class and hence Λ˜n h → P0 δh(y)/M0 (y) = Λ0 h, uniformly in h ranging over a Glivenko–Cantelli class. ˆ Λ) ˆ is bigger than the log likelihood The log likelihood evaluated at (θ, ˜ The point masses Λ{Yi } in the likelihood when evaluated evaluated at (θ0 , Λ). ˆ n,0 and M0 . Specifically we at Λˆ and Λ˜ can be reexpressed in the functions M ˆ ˆ ˜ have that Λ/Λ{Yi } = M0 /Mn,0 (Yi ). This yields the equation

M0 ˆ ˆ ˜ (y) ≥ 0. (θˆ − θ0 )Pn zδ − Pn eθz Λ(y) − eθ0 z Λ(y) + Pn δ log ˆ n,0 M

(5.1)

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In the next paragraphs we prove that this implies that for almost all ω in the underlying probability space there exists θ∞ ∈ Θ such that along a subsequence (θˆn , Λˆn ) → (θ∞ , Λ∞ ) and

M0 (θ∞ − θ0 )P0 zδ − P0 eθ∞ z Λ∞ (y) − eθ0 z Λ0 (y) + P0 δ log (y) ≥ 0, (5.2) M∞,0 for M∞,0 (s) = P0 eθ∞ z 1s≤y ,

Λ∞ h = P 0

δh(y) . M∞,0 (y)

The topology on Λ can be taken equal to the uniform norm on [0, τ ]. Noting that M0 /M∞,0 = dΛ∞ /dΛ0 , we recognize (6.4) as the Kullback–Leibler divergence P0 log(pθ∞ ,Λ∞ /pθ0 ,Λ0 ), which is strictly negative by the identifiability of (θ0 , Λ0 ), unless (θ∞ , Λ∞ ) = (θ0 , Λ0 ). This would finish the proof that (θˆn , Λˆn ) → (θ0 , Λ0 ) almost surely. ˆ n,0 are To deduce (6.4) from (7.4) we note first that the functions M ˆ θz ˆ bounded below by Mn,0 (τ ) = Pn e 1y=τ , which is asymptotically bounded away from zero under our assumptions. Therefore the functions (δ, y) → ˆ n,0 (y) are contained in a Glivenko–Cantelli class, almost surely, if h δh(y)/M ˆ n,0 (y)+ ˆ = P0 δh(y)/M ranges over a Glivenko–Cantelli class. It follows that Λh o(1), almost surely, uniformly in h running through a Glivenko–Cantelli class. ˆ ˆ ˆ ˆ ) ≤ Pn eθz Λ(y) = Pn δ, by the likeSecond, we note that Pn eθz 1y=τ Λ(τ ˆ ) is uniformly bounded above, lihood equation with h = 1 and hence Λ(τ eventually, almost surely. By the compactness of Θ, the sequence θˆn converges to a limit θ∞ , at ˆ n,0 (s) → M∞,0 (s), uniformly in s, almost least along subsequences. Then M ˆ ∞,0 (y) = Λ∞ h almost surely, uniformly surely, and hence Λˆn h → P0 δh(y)/M in h running through a Glivenko–Cantelli class with integrable envelope, still along a subsequence. It now suffices to take limits in (7.4). This is done in two steps. We first replace Pn by P0 adding a o(1)-term, which is permitted, because the classes of functions (z, δ) → zδ, (y, z) → eθz Λ(y) and (y, δ) → δ log M0 /M (y) with θ ∈ Θ, Λ ranging over a uniformly bounded set of monotone, cadlag functions and M ranging over monotone, cadlag functions that are bounded away from zero, is Glivenko–Cantelli. The second step ˆ n,0 by their limits, which is justified by the is to replace θˆn , Λˆn , Λ˜n and M dominated convergence theorem. 5.3.3 Trick 3: Mixture models Our first trick already showed that for proving consistency of a maximum likelihood estimator, it may be useful to apply a general result for M -estimators not to the log density, but to a slightly modified function. In models that depend linearly on a parameter belonging to a convex set, there is an even better choice.

5.3 Consistency of M- and Z-estimators

393

Given a kernel p(x|
z) indexed by z ∈ Z and a probability distribution η on Z, let pη (x) = p(x| z) dη(z). Consider proving the nconsistency of the maximum likelihood estimator ηˆ, which maximizes η → i=1 pη (Xi ) over the set of all probability measures. We can use the linearity of this model, by starting from the observation η for every η and that the likelihood is bigger at ηˆ than at ηt = tη + (1 − t)ˆ t ∈ [0, 1]. Differentiating the inequality Pn log pηˆ/pηt ≥ 0 from the right at t = 0 we obtain pη ≤ 1. Pn pηˆ We might try and use this equation for a consistency proof, but the quotients pη /pηˆ may lack integrability, and it is useful to make a second step. Let L: [0, ∞) → R be a nondecreasing function such that t → L(1/t) is convex. Then p 1 p

1 η ˆ η0 Pn L = Pn L ≥L ≥ L(1) = Pn L . pη pη /pηˆ Pn pη /pηˆ pη0 Thus we may use Theorem 5.8 with the choice mη = L(pη /pη0 ). The choice L(t) =

tα − 1 , tα + 1

α ∈ (0, 1],

is attractive, because then L(t) = −L(1/t) is strictly concave. By Jensen’s inequality p pη η Pη0 L ≤ L Pη0 ≤ L(1). pη0 pη0 Unless pη = pη0 almost surely under Pη0 , the first inequality will be strict and hence the left side will be strictly less than the right side. If the set of functions x → p(x| z) where z ranges over Z is Glivenko– Cantelli, then so is its convex hull, the set of all functions pη . The one-element class consisting of the function 1/ph0 is Glivenko–Cantelli and hence so is the class of all functions pη /pη0 and the class of functions L(pη /pη0 ) when η ranges over all probability distributions on (Z, C), by two applications of Theorem 5.7.   We now obtain the consistency of ηˆ for η0 if we can verify that Pη0 L pη /pη0 is strictly bounded away from its maximal value L(1) if η varies over the complement of a ball of radius δ around η0 . This, of course, depends on the metric we choose for the set of mixing distributions. If we choose a metric for which this set is compact, then it suffices to verify that the map η → pη (x) is contin uous for Pη0 -almost every x, because then so is the map η → Pη0 L pη /pη0 , by the dominated convergence theorem. In many examples the weak topology is appropriate, possibly after first compactifying Z. A semi-metric that always works is the induced Hellinger metric, because, for α = 1/2, p η Pη0 L ≤ − 12 h2 (pη , pη0 ). pη0

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5.4 Nuisance Parameters We close this lecture by noting that the preceding theorems have easy extensions to M - and Z-estimators defined in the presence of nuisance parameters. In the case of M -estimators we might be given measurable functions mθ,η : X → R indexed by a parameter of interest θ and a nuisance parameter η. Given an initial estimator ηˆ for η, we consider θˆ maximizing θ → Pn mθ,ˆη . More generally, given an “estimator” ηˆ(θ) for η that may depend on θ, we consider θˆ maximizing θ → Pn mθ,ˆη(θ) . (The latter criterion is sometimes called a profile criterion function.) Both cases are covered if we allow a general random criterion function m ˆ n,θ (x) = m ˆ n,θ (x; X1 , . . . , Xn ). We shall assume that asymptotically the randomness disappears: m ˆ n,θ → mθ for deterministic, measurable functions mθ . Theorem 5.11. Suppose that there exists a Glivenko–Cantelli class F of  ˆ n,θ : θ ∈ Θ} ⊂ F → 1, functions with integrable envelope such that P n {m P 0 for all x, and that there exists a suppose that supθ∈Θ |m ˆ n,θ − mθ |(x) → point θ0 ∈ Θ such that inf θ:d(θ,θ0 )>δ P mθ < P mθ0 for every δ > 0. Then ˆ n,θ0 implies that d(θˆn , θ0 ) → 0 in probability. Pn mn,θˆn ≥ Pn m Proof. For any random sequence θ˜ and every x, the sequence |mn,θ˜ − mθ˜|(x) is bounded by 2F (x) < ∞, for F an envelope function of the class, and converges in probability to zero. This implies that it converges to zero in mean and hence, by Fubini’s theorem and the dominated convergence theorem, EP |mn,θ˜ − mθ˜| → 0. Consequently, the sequence P (mn,θ˜ − mθ˜) converges to zero in probability. Combining this with the Glivenko–Cantelli assumption we obtain that ˆ n,θ˜n − P mθ˜n | converges to zero in probability. the sequence |Pn m The remainder of the proof is similar to the proof of Theorem 5.10.

Notes The proofs of most results on empirical processes given in this lecture and the following ones can be found in the book [41]. This work also contains a reasonable number of references to the huge literature on empirical processes. We do not refer to this here, apart from mentioning that the Saint-Flours notes by Dudley [9] were a major step in developing the abstract theory of empirical processes. Applications of empirical processes to the analysis of M-estimators and Z-estimators were pioneered by Pollard. See [29], [30]. Trick 1 I learned from [4], trick 2 (applied here to the Cox model for the first time) from Susan Murphy (see [24]), and trick 3 from [27].

6. Lecture: Empirical Processes and Normality of Z-Estimators

In this lecture we continue the discussion of empirical processes, now concentrating on the central limit theorem and uniformity in convergence in distribution, and its applications to deriving the asymptotic distribution of Z-estimators.

6.1 Weak Convergence in Metric Spaces Let (Ωn , Un , Pn ) be a sequence of probability spaces and, for each n, let Xn : Ωn → D be an arbitrary map from Ωn into a metric space D. Definition 6.1. The sequence Xn converges in distribution to a Borel mea
sure L on D if and only if E∗ f (Xn ) → f dL for every bounded, continuous function f : D → R. Here the asterisk * denotes outer expectation, and is necessary because we have not assumed that the maps Xn are Borel measurable. It is defined as   E∗ f (X) = inf EU : U : Ω → R, measurable, U ≥ f (X), EU exists . If X is a Borel measurable map in D, defined on some probability space, with law L, then we also write Xn
X instead of Xn
L. The limit is always assumed to be Borel measurable. Even though the Xn and X are ordinary maps, we also refer to them as “random elements”, as they are defined on a probability space and hence induce randomness on D. In the following, we do not stress the measurability issues. However, we write stars, when necessary, as a reminder that there are measurability issues that need to be taken care of. Although Ωn may depend on n, we do not let this show up in the notation for E∗ and P∗ . Next consider convergence in probability and almost surely. Definition 6.2. An arbitrary sequence of maps Xn : Ωn → D converges in   probability to X if P∗ d(Xn , X) > ε → 0 for all ε > 0. This is denoted by P Xn → X.

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Definition 6.3. An arbitrary sequence of maps Xn : Ωn → D converges almost surely to X if there exists a sequence of (measurable) random variables as ∆n such that d(Xn , X) ≤ ∆n and ∆n → 0 This is denoted by Xn as∗ → X. These definitions also do not require the Xn to be Borel measurable. In the definition of “convergence of probability” we added a star, for “outer probability”. Similar to outer expectation, we define outer probability by   P∗ (X ∈ B) = inf P(A): A ∈ A, A ⊃ X −1 (B) . The definition of “almost sure convergence” is unpleasantly complicated. This cannot be avoided easily, because, even for Borel measurable maps Xn and X, the distance d(Xn , X) need not be a random variable. Most of the well-known properties and relationships of these modes of convergence remain valid under the generalized definitions. We collect the most important ones in the following theorem. Theorem 6.4. For arbitrary maps Xn , Yn : Ωn → D and every random element X with values in D, P X implies Xn
X; (i) Xn → P (ii) Xn → c for a constant c if and only if Xn
c; (iii) if Xn
X, then φ(Xn )
φ(X) for every map φ: D → E that is continuous at every point of a set D0 such that P(X ∈ D0 ) = 1 and such that φ(X) is Borel measurable; P (iv) if Xn
X and d(Xn , Yn ) → 0, then Yn
X; P (v) if Xn
X and Yn → c for a constant c, then (Xn , Yn )
(X, c); P P P (vi) if Xn → X and Yn → Y , then (Xn , Yn ) → (X, Y ).

The metric spaces we are mostly interested in are, besides the Euclidean spaces, spaces of bounded functions equipped with the uniform norm. Given an arbitrary set T let ∞ (T ) be the collection of all bounded functions z: T → R. This is a Banach space under the uniform norm   zT = supz(t). t∈T

Most of the random elements X with values in ∞ (T ) of interest to us are stochastic processes in that their coordinate values Xt = πt ◦ X, for π: ∞ (T ) → R the coordinate projection z → z(t), are random variables. However, many of them are not Borel measurable in ∞ (T ) and hence the preceding extensions of the usual definitions are useful. Earlier extensions based on the σ-field generated by the closed balls, initiated by Dudley and expounded by Pollard, are special cases of the present approach, which is due to Hoffmann–Jørgensen. We use the space ∞ (T ) for defining “uniform weak convergence” of stochastic processes, such as the empirical processes. The next theorem gives

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397

a characterization of weak convergence in this space by finite approximation. It is required that, for any ε > 0, the index set T can be partitioned into finitely many sets T1 , . . . , Tk such that (asymptotically) the variation of the sample paths t → Xn,t is less than ε on every one of the sets Ti , with large probability. Then the behaviour of the process can be described,  error margin, by the behaviour of the marginal vec within a small tors Xn,t1 , . . . , Xn,tk for arbitrary fixed points ti ∈ Ti . If these marginals converge, then the processes converge. Theorem 6.5. A sequence of arbitrary maps Xn : Ωn → ∞ (T ) converges weakly to a tight random element if and only if both of the following conditions hold:   (i) the sequence Xn,t1 , . . . , Xn,tk converges in distribution in Rk for every finite set of points t1 , . . . , tk in T ; (ii) for every ε, η > 0 there exists a partition of T into finitely many sets T1 , . . . , Tk such that     ∗   lim sup P sup sup Xn,s − Xn,t ≥ ε ≤ η. n→∞

i

s,t∈Ti

Proof. We only give the proof of the more constructive part, the sufficiency of (i)-(ii). For each natural number m, partition T into sets T1m , . . . , Tkmm as in (ii) corresponding to ε = η = 2−m . Since the probabilities in (ii) decrease if the partition is refined, we can assume without loss of generality that the partitions are successive refinements as m increases. For fixed m define a semimetric ρm on T by ρm (s, t) = 0 when s and t belong to the same partioning set Tjm , and by ρm (s, t) = 1 otherwise. Every ρm -ball of radius 0 < ε < 1 coincides with a partitioning set. In particular, T is totally bounded for ρm , and the ρm -diameter of a set Tjm is zero. By the nesting of the ∞ (s, t). Then ρ is a partitions, ρ1 ≤ ρ2 ≤ · · · . Define ρ(s, t) = m=1 2−m ρm semimetric such that the ρ-diameter of Tjm is smaller than k>m 2−k = 2−m , and hence T is totally bounded for ρ. Let T0 be the countable ρ-dense subset m constructed by choosing an arbitrary point tm j from every Tj . By assumption (i) and Kolmogorov’s consistency theorem we can construct a stochastic {Xt : t ∈ T0 } on some probability space such that  process   Xn,t1 , . . . , Xn,tk
Xt1 , . . . , Xtk for every finite set of points t1 , . . . , tk in T0 . By the portmanteau lemma and assumption (ii), for every finite set S ⊂ T0 ,   P sup sup |Xs − Xt | > 2−m j

s,t∈Tjm s,t∈S

≤ 2−m .

By the monotone convergence theorem this remains true if S is replaced by T0 . If ρ(s, t) < 2−m , then ρm (s, t) < 1 and hence s and t belong to the same partitioning set Tjm . Consequently, the event in the preceding display with S = T0 contains the event in the following display, and

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 P

sup

ρ(s,t)<2−m s,t∈T0

−m

|Xs − Xt | > 2



≤ 2−m .

This sums to a finite number over m ∈ N. Hence, by the Borel–Cantelli  lemma, for almost all ω, Xs (ω) − Xt (ω) ≤ 2−m for all ρ(s, t) < 2−m and all sufficiently large m. This implies that almost all sample paths of {Xt : t ∈ T0 } are contained in U C(T0 , ρ). Extend the process by continuity to a process {Xt : t ∈ T } with almost all sample paths in U C(T, ρ). Define πm : T → T as the map that maps every partioning set Tjm onto m the point tm j ∈ Tj . Then, by the uniform continuity of X, and the fact that the ρ-diameter of Tjm is smaller than 2−m , X ◦ πm
X in ∞ (T ) as m → ∞   (even almost surely). The processes Xn ◦ πm (t): t ∈ T are essentially km dimensional vectors. By (i), Xn ◦ πm
X ◦ πm in ∞ (T ) as n → ∞, for every fixed m. Consequently, for every Lipschitz function f : ∞ (T ) → [0, 1], E∗ f (Xn ◦ πm ) → Ef (X) as n → ∞, followed by m → ∞. Conclude that, for every ε > 0,  ∗    E f (Xn ) − Ef (X) ≤ E∗ f (Xn ) − E∗ f (Xn ◦ πm ) + o(1)

≤ f lip ε + P∗ Xn − Xn ◦ πm T > ε + o(1). For ε = 2−m this is bounded by f lip 2−m + 2−m + o(1), by the construction of the partitions. The proof is complete.

In the course of the proof of the preceding theorem a semimetric ρ is constructed such that the weak limit X has uniformly ρ-continuous sample paths, and such that (T, ρ) is totally bounded. This is surprising: even though we are discussing stochastic processes with values in the very large space ∞ (T ), the limit is concentrated on a much smaller space of continuous functions. Actually, this is a consequence of imposing the condition (ii), which can be shown to be equivalent to asymptotic tightness. (A sequence Xn is called asymptotically tight if for every ε > 0 there exists a compact set K ⊂ D such that lim inf P(d(Xn , K) < η) ≥ 1 − ε for every η > 0.) It can be shown, more generally, that every tight random element X in ∞ (T ) necessarily concentrates on U C(T, ρ) for some semimetric ρ (depending on X) that makes T totally bounded. In view of this connection between the partitioning condition (ii), continuity and tightness, we sometimes refer to this condition as the condition of asymptotic tightness or asymptotic equi-continuity. One consequence of this is that a tight random element X is completely determined by its values on a countable set (taken dense in (T, ρ)), and hence its distribution is determined by the distributions of all its finite-dimensional projections. The existence of a semi-metric that induces continuity will enable us to use empirical process theory in the analysis of Z-estimators. Thus we record the existence of the semimetric for later reference. We also note that, for

6.2 Donsker Classes

399

a Gaussian limit process, this can always be taken equal to the “intrinsic” standard deviation semimetric. This will help a good deal to make our results on Z-estimators more concrete. Lemma 6.6. Under the conditions (i)–(ii) of the preceding theorem there exists a semimetric ρ on T for which T is totally bounded, and such that the weak limit of the sequence Xn can be constructed to have almost all sample paths in U C(T, ρ). Furthermore, if the weak limit X is zero-mean Gaussian, then this semimetric can be taken equal to ρ(s, t) = sd(Xs − Xt ). Proof. A semimetric ρ is constructed explicitly in the proof of the preceding theorem. It suffices to prove the statement concerning Gaussian limits X. Let ρ be the semimetric obtained in the proof of the theorem and let ρ2 be the standard deviation semimetric. Since every uniformly ρ-continuous function has a unique continuous extension to the ρ-completion of T , which is compact, it is no loss of generality to assume that T is ρ-compact. Furthermore, assume that every sample path of X is ρ-continuous. An arbitrary sequence tn in T has a ρ-converging subsequence tn → t. By the ρ-continuity of the sample paths, Xtn → Xt almost surely. Since every Xt is Gaussian, this implies convergence of means and variances, whence ρ2 (tn , t)2 = E(Xtn − Xt )2 → 0. Thus tn → t also for ρ2 and hence T is ρ2 -compact. Then there Suppose that a sample path t → Xt (ω) is not ρ2 -continuous.   exists an ε > 0 and a t ∈ T such that ρ2 (tn , t) → 0, but Xtn (ω) − Xt (ω) ≥ ε for every n. By the ρ-compactness and continuity, there exists a subsequence such that ρ(tn , s) → 0 and Xtn (ω) → Xs (ω) for some s. By the argument  of the preceding paragraph, ρ2 (tn , s) → 0, so that ρ2 (s, t) = 0 and Xs (ω) − Xt (ω) ≥ ε. Conclude that the path t → Xt (ω) can only fail to be ρ2 continuous for ω for which there exist s, t ∈ T with ρ2 (s, t) = 0, but Xs (ω) = Xt (ω). Let N be the set of ω for which there do exist such s, t. Take a countable, ρ-dense subset A of {(s, t) ∈ T × T : ρ2 (s, t) = 0}. Since t → Xt (ω) is ρ-continuous, N is also the set of all ω such that there exist (s, t) ∈ A with Xs (ω) = Xt (ω). From the definition of ρ2 , it is clear that for every fixed (s, t), the set of ω such that Xs (ω) = Xt (ω) is a nullset. Conclude that N is

a null set. Hence, almost all paths of X are ρ2 -continuous.

6.2 Donsker Classes Given a random sample X1 , . . . , Xn from a probability distribution P on a√ measurable space (X , A), let again Gn be the empirical process Gn f = n(Pn f − P f ) indexed by a given class F of measurable functions. Under the assumptions that the class possesses a finite envelope F and that P F is finite (in particular if P F < ∞) the sample paths f → Gn f are contained in the space ∞ (F).

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Definition 6.7. A class F of functions is P -Donsker if the sequence of empirical processes converges in distribution to a tight limit process in the space ∞ (F). The convergence of the process in ∞ (F) implies the convergence of the  marginals Gn f1 , . . . , Gn fk for given any finite set elements fi ∈ F, by the continuous mapping theorem. This is possible only if P fi2 < ∞ for every i and then the limit distribution is multivariate normal with mean zero and covariances P (fi − P fi )(fj − P fj ) by the multivariate central limit theorem. Thus if F is Donsker, then Gn
G for a tight Gaussian random element in ∞ (F) with mean zero and covariance function EGP f GP g = P f g − P f P g. This is known as a P -Brownian bridge. In view of the results of the preceding section this is also determined by: – G is a Gaussian process; – EGf = 0, cov(Gf, Gg) = P f g − P f P g; – the sample paths of G can be constructed to be uniformly continuous relative to the semimetric ρ(f, g) = sd(Gf − Gg). – F is totally bounded under ρ. The L2 (P )-metric is slightly stronger than the metric ρ, because  2 ρ2 (f, g) = P (f − P f ) − (g − P g) ≤ P (f − g)2 . Thus the sample paths are also uniformly continuous relative to the L2 (P )semimetric. It is not hard to see that F will also be totally bounded relative to the L2 (P )-semimetric as soon as P F < ∞. Thus there is not much loss in replacing ρ by the L2 (P )-metric and for this reason we shall work with the simpler L2 (P )-metric from now on. Just as for the Glivenko–Cantelli theorem, there are two basic theorems that imply that a class of functions is Donsker, using bracketing or covering numbers. It is required that the numbers     or sup N εF Q,r , F, L2 (Q) N[] ε, F, L2 (P ) Q

do not grow too fast as ε ↓ 0. The rate of growth is elegantly measured through the bracketing integral and the uniform entropy integral defined as   J[] δ, F, L2 (P ) =



δ



  log N[ ] ε, F, L2 (P ) dε,

δ



  log sup N εF Q,2 , F, L2 (Q) dε.

0





J δ, F, L2 ) = 0

Q

6.2 Donsker Classes

401

The convergence of
these integrals depends only on the size of the integrands 1 as ε ↓ 0. Because 0 ε−r dε converges for r < 1 and diverges for r ≥ 1, convergence of the integrals roughly requires that the entropies grow at slower order than (1/ε)2 . Theorem theorem). Every class F of measurable functions  6.8 (Donsker  with J[] 1, F, L2 (P ) < ∞ is P -Donsker. Theorem 6.9 (Donsker theorem). Every suitably measurable class F of measurable functions with J(1, F, L2 ) < ∞ and P ∗ F 2 < ∞ is P -Donsker. The condition that the class F be “suitably measurable” is satisfied in most examples, but cannot be omitted. We do not give a general definition here, but note that it suffices that there exists a countable collection G of functions such that each f is the pointwise limit of a sequence gm in G. We shall call a class with this property separable. As remarked in the preceding lecture, many estimates of the bracketing or uniform entropy are available in the literature and can be used to derive concrete Donsker classes. Alternatively, new Donsker classes can be constructed out of known Donsker classes. The following theorem is in this spirit and will be useful. For ease of terminology we call a collection of measurable functions f : X → Rk Donsker if each of the k collections of coordinate functions is Donsker. Theorem 6.10. If F is a Donsker class of functions f : X → Rk with squareintegrable envelope, and φ: Rk → R is Lipschitz, then the class of functions φ ◦ f : X → R is Donsker provided that it has a square-integrable envelope. Our result on Z-estimators should cover the classical results, which are obtained by Taylor expansions. This concerns classes of functions ψθ : X → Rk , where θ ranges over a bounded subset of Rk and the dependence θ → ψθ is “smooth”. The following lemma gives a bound on the entropy of such a class, which shows that these classes are very easily Donsker. Lemma 6.11 (Parametric class). Let F = {fθ : θ ∈ Θ} be a collection of measurable functions indexed by a bounded subset Θ ⊂ Rd . Suppose that there exists a measurable function m such that   fθ (x) − fθ (x) ≤ m(x)θ1 − θ2 , every θ1 , θ2 . 1 2 If P |m|r < ∞, then there exists a constant K, depending on Θ and d only, such that the bracketing numbers satisfy diam Θ d   N[] εmP,r , F, Lr (P ) ≤ K , ε

every 0 < ε < diam Θ.

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6 Lecture: Empirical Processes and Normality of Z-Estimators

Proof. We use brackets of the type [fθ − εm, fθ + εm] for θ ranging over a suitably chosen subset of Θ. These brackets have Lr (P )-size 2εmP,r . If θ ranges over a grid of meshwidth ε over Θ, then the brackets cover F, since, by the Lipschitz condition, fθ1 − εm ≤ fθ2 ≤ fθ1 + εm if θ1 − θ2  ≤ ε. Thus, we need as many brackets as we need balls of radius ε/2 to cover Θ. The size of Θ in every fixed dimension is at most diam Θ. We can cover Θ with fewer than (diam Θ/ε)d cubes of size ε. The circumscribed balls have radius a multiple of ε and also cover Θ. If we replace the centers of these balls by their projections into Θ, then the balls of twice the radius still cover Θ.

6.3 Maximal Inequalities We do not include the proofs of the two Donsker theorems here, but we do include the basic maximal inequalities, on which the proofs rest. These are bounds on the distribution of the supremum variables Gn F . For our main purpose inequalities on the L1 -norm of these variables are sufficient. We use these inequalities in the next section to treat empirical processes indexed by random functions. Actually, the Theorem 6.15 obtained there can easily be turned into a proof of the Donsker theorems. Lemma 6.12. For any class F of measurable  functions f : X → R such that P f 2 < δ 2 for every f , we have, with a(δ) = δ/ Log N[] (δ, F, L2 (P )),   √   √ E∗P Gn F  J[] δ, F, L2 (P ) + nP ∗ F F > na(δ) . Corollary 6.13. For any class F of measurable functions with envelope function F ,     E∗P Gn F  J[] F P,2 , F, L2 (P ) . Proof. Since F is contained in the single bracket [−F, F ], the bracketing  number N[] δ, F, L2 (P ) can be taken equal to 1 for δ = 2F P,2 . Then the constant a(δ) in√the preceding lemma reduces to a multiple   √ as defined of F P,2 , and nP ∗ F F > na(δ) is bounded above by a multiple of F P,2 , by Markov’s inequality.qed Lemma 6.14. For any suitably measurable class F of measurable functions f : X → R, we have, with θn2 = supf ∈F Pn f 2 /Pn F 2 , 

E∗P Gn F  E J(θn , F, L2 )F Pn ,2  J(1, F, L2 )F P,2 .

6.4 Random Functions In Lecture 10 we shall use the preceding theorems directly to ensure that certain stochastic processes appearing in the asymptotic analysis of Z-estimators

6.4 Random Functions

403

converge in distribution. However, our main use for Donsker classes in these lectures is indirect: they give a tool to show study averages of “random functions”. Here by “random functions” we mean measurable functions x → fˆn (x; X1 , . . . , Xn ) that, for every fixed x, are functions of the observations. We write fˆn for fˆn (·; X1 , . . . , Xn ) and use the notations Pn fˆn and P fˆn as abbreviations for the expectations of the functions x → fˆn (x; X1 , . . . , Xn ) with X1 , . . . , Xn fixed. Thus

1 ˆ Gn fˆn = √ fn (Xi ; X1 , . . . , Xn ) − P fˆn , n i=1  P fˆn = fˆn (x; X1 , . . . , Xn ) dP (x). n

Note that Gn fˆn is not centered at mean zero, although it could be considered centered in a wide sense. Obviously, the central limit theorem does not apply to a sequence of the form Gn fˆn . However, if the functions fˆn are sufficiently stable, then its result is still true. Theorem 6.15. If there exists a P -Donsker class F such that P n (fˆn ∈ F) → 1 and P (fˆn − f0 )2 → 0 in probability, for some f0 ∈ L2 (P ), then Gn (fˆn − f0 ) → 0 in probability. Proof. Assume without of loss of generality that f0 is contained in F. Define a function g: ∞ (F) × F → R by g(z, f ) = z(f ) − z(f0 ). The set F is a semimetric space relative to the L2 (P )-metric. The function g is continuous with respect to the product semimetric at every point (z, f ) such that f → z(f ) is continuous. Indeed, if (zn , fn ) → (z, f ) in the space ∞ (F) × F, then zn → z uniformly and hence zn (fn ) = z(fn ) + o(1) → z(f ) if z is continuous at f . P By assumption, fˆn → f0 as maps in the metric space F. Since F is Donsker, Gn
GP in the space ∞ (F), and it follows that (Gn , fˆn )
(GP , f0 ) in the space ∞ (F) × F. By Lemma 6.6, almost all sample paths of GP are continuous on F. Thus the function g is continuous at almost every point (GP , f0 ). By the continuous mapping theorem, Gn (fˆn − f0 ) = g(Gn , fˆn )
g(GP , f0 ) = 0. The lemma follows, since convergence in distribution and convergence in probability are the same for a degenerate limit.

Employing a fixed Donsker class in the preceding lemma gives a useful, relatively simple condition for getting rid of randomness in the function fˆn . The lemma covers many examples. However, other methods may give better results. Sometimes it is possible to study Gn (fˆn − f0 ) by direct methods, such as computing means and variances. In other situations it is good to know that what is really needed is not that the functions fˆn remain within

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a fixed class, as n → ∞, but that the complexity of the set of functions fˆn does not increase too much with n. We can make this precise through a formulation using entropy conditions. On the one hand this gives more flexibility. On the other hand, nice results such as Theorem 6.10, which allow a calculus to create new Donsker classes, become unavailable. In the next theorem we require that the realizations of the random functions fˆn belong to classes Fn that may change with n. We assume that these classes possess envelope functions Fn that satisfy the Lindeberg condition P Fn2 = O(1), √ P Fn2 {Fn > ε n} → 0, every ε > 0. Then the result of the preceding theorem remains true provided the entropy integrals of the classes behave well. Theorem 6.16. Let Fn be classes of measurable functions such that P n (fˆn ∈ Fn ) → 1 and such that either   (i) J[] δn , Fn , L2 (P ) → 0 or (ii) J(δn , Fn , L2 ) → 0, for every δn ↓ 0, and with envelope functions that satisfy the Lindeberg condition. In the case of (ii) also assume that the classes are suitably measurable. If P (fˆn −f0 )2 → 0 in probability for some f0 ∈ L2 (P ), then Gn (fˆn − f0 ) → 0. Proof. Without loss of generality assume that f0 = 0. Otherwise, replace Fn by Fn − f0 and fˆn by fˆn − f0 . {f ∈ First assume that (i) holds. Let Gn (δ) be the set of functions  2 2 n ˆ Fn : P f ≤ δ } By assumption we have that P fn ∈ Gn (δ) → 1 as n → ∞, for every δ > 0. On the event {fˆn ∈ Gn (δ)} we have |Gn fˆn | ≤ supg∈Gn (δ) |Gn g|. By Lemma 6.12  √    P Fn2 1 Fn > an (δ) n ∗ , E sup |Gn g|  J[] δ, Gn (δ), L2 (P ) + an (δ) g∈Gn (δ) where an (δ) is the number given in Lemma 6.12 evaluated for the class of functions Gn (δ). The first term on the right increases  if we replaceGn (δ) by δn , Fn , L2 (P ) → 0 for J Fn and hence converges to zero as δ → 0. Since []  every δn ↓ 0, we must have that J[] δ, Fn , L2 (P ) = O(1) for every δ > 0 and hence      δ log N[] δ, Fn , L2 (P ) ≤ J[] δ, Fn , L2 (P ) = O(1). Therefore, an (δ) is bounded away√from   zero, for every fixed δ as n → ∞. Conclude that P Fn2 1 Fn > an (δ) n → 0 as n → ∞ followed by δ → 0. The proof under (i) is complete. Next assume that (ii) holds. The class Gn (δ), defined as before, has envelope function 1 + Fn and hence, by Lemma 6.14,

6.4 Random Functions

E∗

405

    sup |Gn g|  E∗ J θn (δ), Gn (δ), L2 Pn (1 + Fn )2 , g∈Gn (δ)

for J the uniform entropy integral of Gn (δ) relative to the envelope function 1 + Fn and Pn f 2 Gn (δ) ≤ Pn f 2 Gn (δ) ∧ 1. θn2 (δ) = Pn (1 + Fn )2 The covering numbers of Gn (δ) are bounded by by the covering numbers of Fn and hence the uniform entropy integral of Gn (δ) is bounded by the uniform entropy integral of Fn if we compute them relative to the same envelope function. If for Fn we replace the envelope 1 + Fn by the natural envelop Fn , then the uniform entropy integral increases. Thus we can further bound the right side of (6.4) by 

1/2   E∗ J 2 θn (δ), Fn , L2 E∗ Pn (1 + Fn )2  1/2   1/2  P (1 + Fn )2  J 2 (1, Fn )P∗ θn (δ) ≥ ε + J 2 (ε, Fn , L2 ) .

We conclude that the theorem is proved if we can show that θn (δ) → 0 in probability as n → ∞, followed by δ → 0.  2  √ Fix η > 0. The class √ of functions Hn (δ, η) = f 1Fn ≤η n : f ∈ Gn (δ) has envelope function η nFn . Hence by Lemma 6.14  √  E∗ Gn Hn (δ,η)  J 1, Hn (δ, η), L2 η nFn P,2 . (6.1) Because √ Q(f 2 1Fn ≤η√n − g 2 1Fn ≤η√n )2 ≤ Q(f − g)2 (2η n)2 , we have  √    N εη nFn Q,2 , Hn (δ, η), L2 (Q) ≤ N 12 εFn Q,2 , Fn , L2 (Q) . Inserting this in the right√side of (7.4) we see that the left side of (7.4) is bounded by J(1, Fn , L2 )η nFn P,2 . We conclude that E∗ Pn −P Hn (δ,η) → 0 as n → ∞ followed by η → 0.  For any fixed η > 0 the class of functions Hn (δ, η) = f 2 1Fn >η√n : f ∈  Gn (δ) satisfies E∗ Pn − P Hn (δ,η) ≤ 2P Fn2 1Fn >η√n → 0. Combined with the result of the preceding paragraph this yields E∗ Pn − P Gn (δ)2 → 0, as n → ∞, for every δ > 0. Because also P Gn (δ)2 ≤ δ 2 by the definition of the class Gn (δ), we conclude that Pn Gn (δ)2 → 0 as n → ∞ followed by δ > 0. This concludes the proof.

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6.5 Asymptotic Normality of Z-Estimators ˆ defined as a zero In the preceding lecture we showed that a Z-estimator θ, of a random criterion function θ → Pn ψθ , is typically consistent for a zero of the limiting criterion function θ → P ψθ . The asymptotic distribution of the difference θˆ − θ depends on the fluctuations of the random criterion function Pn ψθ around its limit P ψθ . Empirical processes are what we need to study such fluctuations. We start with a simple theorem. For every θ in a set Θ ⊂ Rk let ψθ : X → k R be a measurable, vector-valued function. Theorem 6.17. Suppose that the class of functions {ψθ : θ ∈ Θ} is P Donsker, that the map θ → P ψθ is differentiable at θ0 with nonsingular derivative V , and that the map θ → ψθ is continuous in L2 (P ) at θ0 . Then P any θˆn such that Pn ψθˆn = 0 and such that θˆn → θ0 for a zero θ0 of θ → P ψθ satisfies √ n(θˆn − θ0 ) = −Vθ−1 Gn ψθ0 + oP (1). 0 Proof. The consistency of θˆn and the Donsker condition on the functions ψθ imply that P 0. (6.2) Gn ψθˆn − Gn ψθ0 → √ By the definitions of θˆn and θ0 , we can rewrite Gn ψθˆn as nP (ψθ0 − ψθˆn ) + oP (1). Combining this with the Delta-method and the differentiability of the map θ → P ψθ , we find that   √ √ nVθ0 (θ0 − θˆn ) + n oP θˆn − θ0  = Gn ψθ0 + oP (1). In particular, by the invertibility of the matrix Vθ0 ,  √  √ √  nθˆn − θ0  ≤ Vθ−1  nVθ0 (θˆn − θ0 ) = OP (1) + oP nθˆn − θ0  . 0 √ This implies that θˆn is n-consistent: the left side is bounded in probabil√ ity. Inserting this in the previous display, we obtain that nVθ0 (θˆn − θ0 ) = left and −Gn ψθ0 + oP (1). We conclude the proof by taking the inverse Vθ−1 0 right. Since matrix multiplication is a continuous map, the inverse of the remainder term still converges to zero in probability.

This theorem as stated covers most (or all?) of the popular examples of Z-estimators, the condition that the functions ψθ form a Donsker class being not at all very restrictive. The Donsker class condition is used to ensure (6.2) and can be relaxed to   √ Gn (ψθˆn − ψθ0 ) = oP 1 + nθˆn − θ0  without changing the remainder of the proof. Of course, this or (6.2) does not really require that the class {ψθ : θ − θ0  < δ} is Donsker for any fixed

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δ, but concerns a limiting property of these classes as δ → 0. Potentially, the Donsker condition could be relaxed to a condition that directly involves entropy numbers. Such a relaxation does not appear to be worth the trouble in the situation of the preceding theorem, but is potentially of use in situations with nuisance parameters or criterion functions that change with n.

6.6 Nuisance parameters An important method of estimation for semiparametric models, but also in general, is Z-estimation in the presence of nuisance parameters. We are given measurable functions ψθ,η : X → Rk indexed by a parameter of interest θ ∈ Rk and a nuisance parameter η belonging to some metric space. Given an initial estimator ηˆ for η, we consider the (near) solution θˆ of the equation Pn ψθ,ˆη = 0.  Theorem 6.18. Suppose that the class of functions ψθ,η : θ − θ0  < δ,  d(η, η0 ) < δ is Donsker for some δ > 0, that the maps θ → P ψθ,η are differentiable at θ0 , uniformly in η in a neighbourhood of η0 with nonsingular and assume that the map derivative matrices Vθ0 ,η such that Vθ0 ,η → Vθ0 ,η0 , √ (θ, η) → ψθ,η is continuous in L2 (P ) at (θ0 , η0 ). If n Pn ψθˆn ,ˆηn = oP (1) and P (θ0 , η0 ) for a point (θ0 , η0 ) satisfying P ψθ ,η = 0, then (θˆn , ηˆn ) → 0

√

0

√ n(θˆn − θ0 ) = − Vθ−1 n P ψθ0 ,ˆηn − Vθ−1 ,η Gn ψθ0 ,η0 (Xi ) 0 ,η0  0 0 √ + oP 1 + nP ψθ0 ,ˆηn  .

Proof. The proof closely follows the proof of the theorem without nuisance parameters. The consistency of (θˆn , ηˆn ) and the Donsker condition imply that P 0. Gn ψθˆn ,ˆηn − Gn ψθ0 ,η0 →

(6.3)

Because (θˆn , ηˆn ) and (θ0 , η0 ) are zeros of the random criterion function and its limit, we can rewrite this as √ −Gn ψθˆ0 ,ˆη0 = nP (ψθˆn ,ˆηn − ψθ0 ,η0 ) + oP (1) . (6.4) √ √ = n(P (ψθˆn ,ˆηn − ψθ0 ,ˆηn ) + nP ψθ0 ,ˆηn + oP (1). By the uniform differentiability of the map θ → P ψθ and the uniform nonsingularity of its derivative, we find that there exists c > 0 such that for all (θ, η) in a sufficiently small neighbourhood of (θ0 , η0 )   P (ψθ,η − ψθ ,η ) ≥ cθ − θ0 . 0 0 Combined with the preceding display this shows that with probability tending to one,

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  √   √ cθˆ − θ0  ≤ Gn ψθˆ0 ,ˆη0  + nP ψθ0 ,ˆηn  = OP 1 + nP ψθ0 ,ˆηn  . We now linearize the first term on the far right of (6.4) in θˆ − θ0 and finish the proof as before.

Under√the conditions of this theorem, the limiting distribution of the sequence n(θˆn − θ0 ) depends on the estimator ηˆn through the “drift” term √ nP ψθ0 ,ˆηn . In general, this gives a contribution to the limiting distribution, √ and ηˆn must be chosen with care. If ηˆn is n-consistent and the map η → P ψθ0 ,η is differentiable, then the drift term can be analyzed using the Deltamethod. It may happen that the drift term is zero. If the parameters θ and η are “orthogonal” in this sense, then the auxiliary estimators ηˆn may converge at an arbitrarily slow rate and affect the limit distribution of θˆn only through their limiting value η0 . In semiparametric situations it is quite common to set up the estimating equations such that the drift term gives a zero contribution. Then the advantage of using a random value ηˆn over a fixed value could be a gain in efficiency: we choose ηˆn to converge to a value η0 such that the asymptotic covariance matrix P ψθ0 ,η0 ψθT0 ,η0 VθT0 ,η0 Vθ−1 0 ,η0 is “small”. This theorem and discussion is valid whether (θ, η) completely parametrizes a model, or not. In the first case, we would write a true distribution Pθ0 ,η0 rather than as P . The asymptotic covariance matrix in the preceding display would then be at least equal to the inverse of the efficient information matrix. It would be equal to this if ψθ,η is proportional to the the efficient score function for θ. Example 6.19 (Regression). Let a typical observation be a pair X = (Y, Z) whose distribution is described structurally by the equation Y = fθ (Z) + e for (Z, e) having a distribution η such that Eη (e| Z) = 0. Consider the estimation equation defined by   ψθ,η (x) = y − fθ (z) wθ,η (z), for given weight functions wθ,η . We have

   Pθ0 ,η0 ψθ0 ,η = Eθ0 ,η0 Eθ0 ,η0 Y − fθ0 (Z) | Z wθ0 ,η (Z) = 0. Thus the drift term in the preceding theorem vanishes. To obtain an efficient estimator we must choose the weight function equal to wθ,η (z) = g˙ θ (z)/Eη (e2 | Z = z) and use estimators for η such that wθ,ˆη is consistent for this weight function, but (almost) any choice of the weight function will work to obtain an asymptotically normal estimator. One explanation for the fact that these estimating equations are unbiased is that the functions belong to the orthocomplement of the tangent set.

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409

In a number of models, such as the regression model in the preceding example, setting up good estimating equations is easy. In general, calculation of the tangent set of a model, or rather its orthocomplement can be of help. First, if some function ψθ,η is orthogonal to the tangent set due to the nuisance parameters, its mean Pθ,η ψθ,ˆη should be fairly insensitive to the estimator ηˆ, because by definition a nuisance score gives the change in the underlying distribution if perturbing the nuisance parameter. One attempt to make this idea formal is to write  p − p θ,ˆ η θ,h Pθ,η ψθ,ˆη = (Pθ,η − Pθ,ˆη )(ψθ,ˆη − ψθ,η ) − Pθ,η − Bθ,η h ψθ,η , (6.5) pθ,η where Bθ,η h can be any η-score if ψθ,η is orthogonal to the nuisance tangent space. If Bθ,η h can approximate (pθ,ˆη − pθ,h )/pθ,η , then we might hope that η , η)2 ), for the metric d the right side of the display is of the order OP (d(ˆ giving the approximation. Then the drift term will give no contribution to the limit distribution if d(ˆ η , η) = oP (n−1/4 ). This informal argument can be useful, but it should not be concluded that a n−1/4 -rate for the nuisance parameter is “minimal” in some sense. Special properties of the model, as in the regression example, may make the drift term zero for any ηˆ. The point is that P ψθ,ˆη is an integrated quantity and it is far to crude to analyse it by a Taylor expansion, replacing the integrand by its absolute value after subtracting the beginning of the expansion. Nevertheless, we can formalize the expansion, for instance, as follows. Given some semiparametric model P = {Pθ,η : θ ∈ Θ, η ∈ H} with H a metric space, suppose that, for some nonnegative numbers α, β, γ,   η , η)2α Pθ,η ψθ,ˆη − ψθ,η 2 = OP d(ˆ 2 p − p   θ,ˆ η θ,η inf η , η)2β+2γ Pθ,η − g = OP d(ˆ ˙ p g∈lin η PPθ,η θ,η  p − p 2   θ,ˆ η θ,η Pθ,η η , η)2β . = OP d(ˆ pθ,η   η , η)δ for δ = (a ∨ γ) + β. Then Pθ,η ψθ,ˆη = OP d(ˆ If the underlying measure P = Pθ,η belongs to a semiparametric model, then it is worth while to adapt the conditions of Theorem 6.18 somewhat and to use the differentiability of the model in θ. This leads to the following theorem, which we shall apply in the next lectures to construct efficient estimators or analyse the maximum likelihood estimator. We now make the disappearance of the bias term part of the conditions. Let ψθ,η : X → R be measurable functions and and let ηˆn be estimators such that   (6.6) Pθˆn ,η ψθˆn ,ˆηn = oP n−1/2 + θˆn − θ ,  2 P  2 (6.7) Pθ,η ψθˆn ,ˆηn − ψθ,η  → 0, Pθˆn ,η ψθˆn ,ˆηn  = OP (1).

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The second condition (6.7) merely requires that the “plug-in” estimator ψθ,ˆηn is a consistent estimator for the “true” estimating function ψθ,η . If Pθ,η ψθ,η = 0, as we shall require, then the first condition (6.6) can be understood as requiring that the “bias” of the plug-in estimator,√due to estimating the nuisance parameter, converges to zero faster than 1/ n. Note that the derivative of θ → Pθ,η ψθ,ˆη should converge to the derivative of θ → Pθ,η ψθ,η , which is zero, and hence, informally the condition (6.6) must be equivalent to √ P 0, . (6.8) nPθ,η ψθ,ˆηn → Theorem 6.20. Suppose that the model {Pθ,η : θ ∈ Θ} is differentiable in quadratic mean with respect to θ at (θ, η). Let the matrix Pθ,η ψθ,η ˙θ,η be nonsingular. Assume that (6.6) and (6.7) hold. Furthermore, suppose that there exists a Donsker class with square-integrable envelope function that contains every function ψθˆn ,ˆηn with probability tending to 1. Then a zero θˆn of √ θ → Pn ψθ,ˆηn that is consistent for θ satisfies that n(θˆn −θ) is asymptotically normal with mean zero and covariance matrix T T −1 (Pθ,η ψθ,η ˙Tθ,η )−1 P ψθ,η ψθ,η (Pθ,η ˙θ,η ψθ,η ) . √ Proof. Let Gn (θ , η  ) = n(Pn − Pθ,η )ψθ ,η be the empirical process indexed by the functions ψθ ,η . By the assumption that the functions ψθ,ˆ ˆ η are contained in a Donsker class, together with (6.7),

Gn (θˆn , ηˆn ) = Gn (θ, η) + oP (1). (Cf. Theorem 6.15.) By the defining relationship of θˆn and the “no-bias” condition (6.6), this is equivalent to   √ √ n(Pθˆn ,η − Pθ,η )ψθˆn ,ˆηn = Gn (θ, η) + oP 1 + nθˆn − θ0  . The remainder of the  proof consists √ of showing that the left side is asymptotically equivalent to V + oP (1) n(θˆn − θ) for V = Pθ,η ψθ,η ˙Tθ,η , from which the theorem follows. The difference of the left side of the preceding display √ and V n(θˆn − θ) can be written as the sum of three terms:  √ 1/2 1/2 1/2 1/2 1/2  n ψθˆn ,ˆηn (pθˆ ,η + pθ,η ) (pθˆ ,η − pθ,η ) − 12 (θˆn − θ)T ˙θ,η pθ,η dµ n n  √ 1/2 1/2 1/2 + ψθˆn ,ˆηn (pθˆ ,η − pθ,η ) 12 ˙Tθ,η pθ,η dµ n(θˆn − θ) n  √ − (ψθˆn ,ˆηn − ψθ,η ) ˙Tθ,η pθ,η dµ n(θˆn − θ). √  The first and third term can easily be seen to be oP nθˆn − θ by applying the Cauchy–Schwarz inequality together with the differentiability of the

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411

model and (6.7). The square of the norm of the integral in the middle term can for every sequence of constants mn → ∞ be bounded by a multiple of  m2n

1/2

1/2

1/2

ψθˆn ,ˆηn  pθ,η |pθˆ ,η − pθ,η | dµ2 n   2 + ψθˆn ,ˆηn  (pθˆn ,η + pθ,η ) dµ

˙θ,η >mn

˙θ,η 2 pθ,η dµ.

In view of (6.7), the differentiability of the model in θ and the Cauchy– Schwarz inequality, the first term converges to zero in probability provided P mn → ∞ sufficiently slowly to ensure that mn θˆn −θ → 0. (Such a sequence  P exists. If Zn → 0, then there exists a sequence εn ↓ 0 such that P |Zn | >  −1/2 P εn → 0. Then εn Zn → 0.) In view of the last part of (6.7), the second term converges to zero in probability for every mn → ∞. This concludes the proof of the theorem.

In the preceding theorems we have assumed that the realizations of the functions ψθ,ˆ ˆ η are contained in a fixed Donsker class, with high probability. This condition is overly strong. As we pointed out in Lecture 6, what is needed is that the entropy of these collections of realizations is asymptotically stable and not too big. Hence the condition can be replaced by the condition that there exist classes Fn of functions satisfying the conditions of Theorem 6.16 such that ψθ,ˆ ˆ η is contained in Fn with probability tending to one. One further extension is to permit ψθ,η to change with n itself. Notes. See the notes to Lecture 5. The general topic of Section 6.4 is taken from [42], but the main result here is new.

7. Lecture: Efficient Score and One-step Estimators

In this lecture we consider the construction of efficient estimators in semiparametric models using the efficient score equation or the related one-step method. We apply it to the linear errors-in-variables model and the symmetric location model.

7.1 Efficient Score Estimators The most important method to estimate the parameter in a parametric model is the method of maximum n likelihood, and it can usually be reduced to solving the score equations i=1 ˙θ (Xi ) = 0, if necessary in a neighbourhood of an initial estimate. A natural generalization to estimating the parameter θ in a semiparametric model {Pθ,η : θ ∈ Θ, η ∈ H} is to solve θ from the efficient score equations n (7.1) ˜θ,ˆηn (Xi ) = 0. i=1

Here we use (a version of) the efficient score function instead of the ordinary score function, and we substitute an estimator ηˆn for the unknown nuisance parameter. Alternatively, it may be more workable to find an “estimator” ηˆn (θ) for η acting as if θ is known already and next solve θ from the “profile efficient score equations” n

˜θ,ˆηn (θ) (Xi ) = 0.

i=1

A solution θˆn also satisfies the efficient score equation (7.1) if we set ηˆn = ηˆn (θˆn ). This choice of ηˆn may beat the purpose of finding an estimator θˆn , but this remark does indicate that to prove something about θˆn it is not necessary to consider the profile efficient score equation. Hence we concentrate on solutions of (7.1). We can derive the asymptotic normality of θˆn from Theorem 6.20. Here Pθ,η ψ˜θ,η ˙Tθ,η is the efficient information matrix I˜θ,η and hence the asymptotic −1 . covariance matrix in this theorem reduces to I˜θ,η

7.2 One-step Estimators

413

Theorem 7.1. Suppose that conditions of Theorem 6.20 are satisfied with ψθ,η = ˜θ,η . Then a consistent sequence of zeros θˆn of θ → Pn ˜θ,ˆηn is asymptotically efficient for ψ(Pθ,η ) = θ at (θ, η). We remark again that the condition that the functions ˜θ,η are contained in a fixed Donsker class can be relaxed along the lines of Theorem 6.16.

7.2 One-step Estimators Theorem 7.1 applies to many examples, but its conditions are not the minimal ones to ensure existence of asymptotically efficient estimators. There are may ways in which its conditions can be relaxed, all leading to estimators that are less natural but have better properties, in theory. We shall immediately go to the most extreme modification, which can be shown to work whenever there is anything that works. √ Suppose that we are given a sequence of initial estimators θ˜n that is nconsistent for θ. We can assume without loss of generality that the estimators are discretized on a grid of mesh width n−1/2 , which will simplify the constructions and proof. Then the one-step estimator is defined as θˆn = θ˜n −

n

n

−1

ˆn,θ˜n ,i ˆTn,θ˜ ,i (Xi ) n

i=1

ˆn,θ˜n ,i (Xi ),

i=1

where ˆn,θ,i is an estimator for ˜θ,η . The estimator θˆn can be considered a onestep iteration of  the Newton-Raphson algorithm for solving an approximation to the equation ˜θ,η (Xi ) = 0 with respect to θ, starting at the initial guess θ˜n . For the benefit of the simple proof, we have made the estimators ˆn,θ,i for the efficient score function dependent on the index i. In fact, we shall use only two different values for ˆn,θ,i , one for the first half of the sample, and another for the second half. Given estimators ˆn,θ = ˆn,θ (·; X1 , . . . , Xn ) define, with m = n/2,  if i > m ˆm,θ (·; X1 , . . . , Xm ) ˆ n,θ,i = ˆ n−m,θ (·; Xm+1 , . . . , Xn ) if i ≤ m. Thus, for Xi belonging to the first half of the sample, we use an estimator ˆn,θ,i based on the second half of the sample, and vice versa. This samplesplitting trick is convenient in the proof, because the estimator “of η” used in ˆn,θ,i is always independent of Xi , simultaneously for Xi running through each of the two halves of the sample. The trick is not recommended in practice. The conditions of the preceding theorem can now be relaxed in two ways: we can drop the Donsker condition and we need an analogue of the “no-bias” condition (6.6) only for deterministic sequences θn . We assume that, for every deterministic sequence θn = θ + O(n−1/2 ),

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√

 2 P P n,θn − ˜θn ,η  → nPθn ,η ˆn,θn → 0, Pθn ,η ˆ 0.   2 ˜ 1/2 1/2  θn ,η dPθn ,η − ˜θ,η dPθ,η  → 0.

(7.2) (7.3)

Theorem 7.2. Suppose that the model {Pθ,η : θ ∈ Θ} is differentiable in quadratic mean with respect to θ at (θ, η), let the efficient information matrix I˜θ,η be nonsingular. Assume that (7.2) and (7.3) hold. Then the sequence θˆn is asymptotically efficient at (θ, η). Proof. Fix a deterministic sequence of vectors θn = θ + O(n−1/2 ). By the ˆ sample-splitting, the first half of the sum n,θn ,i (Xi ) is a sum of conditionally independent terms, given the second half of the sample. Thus, √

√   Eθn ,η mPm ˆn,θn ,i − ˜θn ,η | Xm+1 , . . . , Xn = mPθn ,η ˆn,θn ,i , √

 2   n,θn ,i − ˜θn ,η  . varθn ,η mPm ˆn,θn ,i − ˜θn ,η | Xm+1 , . . . , Xn ≤ Pθn ,η ˆ Both expressions converge to zero in probability by assumption (7.2). We conclude that the sum inside the conditional expectations converges conditionally, and hence also unconditionally, to zero in probability. By symmetry, the same is true for the second half of the sample, whence   P √ 0. nPn ˆn,θn ,i − ˜θn ,η → We have proved this for the probability under (θn , η), but by contiguity the convergence is also under (θ, η). Combining the preceding display with the result of Lemma 7.3, we find that   √ √ P 0. nPn ˆn,θ ,i − ˜θ,η + I˜θ,η n(θn − θ) → n

In view of the discretised nature of θ˜n , this remains true if the deterministic sequence √ θn is replaced by θ˜n . This follows, because, for a given M , on the event { nθ˜n − θ ≤ M } the estimator θ˜n can take on only finitely many values, with the total number of different values being bounded independent of n. Thus an expression of the type Gn (θ˜n ) can be bounded above by supθn Gn (θn ) for the supremum ranging over a finite number of points. If each of the sequences Gn (θn ) converges to zero√in probability, then Gn (θ˜n ) converges to zero the event { nθ˜n − θ ≤ M }. Finally, by √ in probability on ˜ the assumed n-consistently of θn , we can fix M such that the probability of this event is arbitrarily close to 1. Next we study the estimator for the information matrix. For any vector h ∈ Rk , the triangle inequality yields  2     Pm (hT ˆn,θ ,i )2 − Pm (hT ˜θ ,η )2  ≤ Pm (hT ˆn,θ ,i − hT ˜θ ,η )2 . n n n n  

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415

By (7.2), the conditional expectation under (θn , η) of the right side given Xm+1 , . . . , Xn converges in probability to zero. A similar statement is valid for the second half of the observations. Combining this with (7.3) and the law of large numbers, we see that P ˜ Pn ˆn,θn ,i ˆTn,θn ,i → Iθ,η .

In view of the discretised nature of θ˜n , this remains true if the deterministic sequence θn is replaced by θ˜n . The theorem follows upon combining the results of the last two paragraphs

with the definition of θˆn . Lemma 7.3. Suppose that the model {Pθ,η : θ ∈ Θ} is differentiable in quadratic mean with respect to θ at (θ, η), let the efficient information matrix I˜θ,η be nonsingular, and assume that (7.3) holds. Then, for any θn = θ + O(n−1/2 ), √

  √ P 0. nPn ˜θn ,η − ˜θ,η + nI˜θ,η (θn − θ) →

Proof. By the definition of the efficient score function as an orthogonal projection, Pθ,η ˜θ,η ˙Tθ,η = I˜θ,η . We shall use this identity several times in the following proof. The lemma follows from adding the two assertions √

nPn

√

1/2 pθn ,η

1 √ P ˜ 0 θn ,η 1 − 1/2 + 2 I˜θ,η n(θn − θ) → pθ,η



nPn ˜θn ,η

1/2

pθn ,η 1/2

pθ,η

√

(7.4)

P 0. − ˜θ,η + 12 I˜θ,η n(θn − θ) →

For the second assertion we note that the variance of the variable on the left side under (θ, η) converges to zero by (7.3). Furthermore, the mean of this variable is equal to   √ √ 1/2 1/2 1/2 1/2 1/2 n ˜θn ,η pθn ,η pθ,η dµ = n ˜θn ,η pθn ,η (pθ,η − pθn ,η ) dµ. √ This is asymptotically equivalent to − 12 nI˜θ,η (θn − θ) by (7.3), the differentiability of the model and the continuity of the inner product. We prove the first assertion in (7.4) also by computing moments, but this time under the measures obtained by letting X1 , . . . , Xn be an i.i.d. sample 1/2 1/2 from the probability measure with density qn = cn pθn ,η pθ,η , where cn is the norming constant. By the differentiability of the model we have   1/2 1/2 1/2 1/2 1 (pθn ,η − pθ,η )2 dµ c−1 = p p dµ = 1 − n θn ,η θ,η 2

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= 1 − 12 (θn − θ)T Iθ,η (θn − θ) + o(n−1 ). From an expansion of the log likelihood ratio of the n-fold product measure n Qnn corresponding to qn and the n-fold product Pθ,η , we see that these product measures are contiguous. Thus it suffices to prove convergence in probability to zero under Qnn . We have 1/2  √ pθn ,η

√ 1   Qnn  nPn ˜θn ,η 1 − 1/2 + n 2 Pn ˜θn ,η ˙Tθ,η (θn − θ) pθ,η    1/2 √  1/2 1/2 1/2  ≤ cn |˜θn ,η pθn ,η | n(pθn ,η − pθ,η ) − 12 (θn − θ)T ˙θ,η pθ,η  dµ → 0,

by the differentiability of the model, (7.3) and the fact that cn → 1. Finally, it suffices to show that the sequence Pn ˜θn ,η ˙Tθ,η converges in probability to I˜θ,η under Qnn . For this we first note that  EQn Pn ˜θn ,η ˙Tθ,η = cn

1/2 1/2 ˜θn ,η pθn ,η ˙Tθ,η pθ,η dµ → I˜θ,η ,

varQn Pn ˜θn ,η 1˜θ,η ≤M ˙Tθ,η 1˙θ,η ≤M ≤ cn M 2

1 n

 1/2 1/2 ˜θn ,η pθn ,η ˙θ,η pθ,η dµ → 0,

for every fixed M . We also have that EQn Pn ˜θn ,η 1˜θ,η >M ˙Tθ,η 1˙θ,η >M → 0, as n → ∞, followed by M → ∞. The proof is complete upon combining the last two displays.

The theorems reduce the problem of efficient estimation of θ to estimation of the efficient score function. At first sight we have made the problem harder. The estimator of the efficient score function must satisfy a “no-bias” and a consistency condition. The consistency is usually easy to arrange, but the nobias condition, such as (7.2), is connected to the structure and the√size of the model, as the bias must converge to zero at a rate faster than 1/ n. It may happen that the bias is identically zero and then we only need to produce a consistent estimator of the efficient score function. In general, we can at best hope that the bias is a second order term, just as in our discussion of general estimating equations in Lecture 6. The good news is that if an efficient estimator sequence exists, then it can always be constructed by the one-step method. In that sense the no-bias condition is necessary.

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Theorem 7.4. Suppose that the model {Pθ,η : θ ∈ Θ} is differentiable in quadratic mean with respect to θ at (θ, η), let the efficient information matrix I˜θ,η be nonsingular, and assume that (7.3) holds. Then the existence of an asymptotically efficient sequence of estimators of ψ(Pθ,η ) = θ implies the existence of a sequence of estimators ˆn,θ satisfying (7.2). Proof. An efficient estimator sequence Tn must be asymptotically linear in the efficient influence function. By Lemma 7.3 and the continuity of θ → I˜θ,η this implies that, for every θn = θ + O(n−1/2 ), √ n(Tn − θn ) = Gn ψ˜θn ,η + oP (1), −1 ˜ θ,η . For simplicity we assume that this expansion is actually where ψ˜θ,η = Iθ,η true in the stronger sense that, for every θn = θ + O(n−1/2 ),

Eθn ,η

√

n(Tn − θn ) − Gn ψ˜θn ,η

2

→ 0.

The general case can be handled by a truncation argument, which turns convergence in probability in convergence in second mean. (See [14].) Furthermore, to simplify notation we assume that Tn is permutation symmetric in its arguments. In view of H´ ajek’s projectionlemma (which gives the orthogonal projecn tion onto the space of all sums i=1 f (Xi )), our assumption implies that Eθn ,η

n 

Eθn ,η

√

2  √ n(Tn − θn )| Xi − Eθn ,η n(Tn − θn ) − Gn ψ˜θn ,η ) → 0,

i=1

which can be rewritten as  2   Eθn ,η Eθn ,η n(Tn − θn )| X1 − Eθn ,η n(Tn − θn ) − ψ˜θn ,η (X1 ) → 0. ˜ Rather  than estimate ψθ,η we can therefore “estimate” the function x → Eθ,η n(Tn − θ)| X1 = x and its expectation. Given kn independent copies Yj1 , . . . , Yjn of the sample X1 , . . . , Xn , define Jn (x) =

kn   1 n Tn (x, Yj2 , . . . , Yjn ) − Tn (Yj1 , . . . , Yjn ) . kn j=1

   Then Eθn ,η Jn (X1 )| X1 ) is identical to Eθn ,η n(Tn −θn )| X1 −Eθn ,η n(Tn −θn ) and hence  2   Eθn ,η Jn (X1 ) − Eθn ,η n(Tn − θn )| X1 + Eθn ,η n(Tn − θn ) =

 2 1 n Eθn ,η n2 Tn (X1 , Yj2 , . . . , Yjn ) − Tn (Yj1 , . . . , Yjn )  , kn kn

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because nEθn ,η (Tn − θn )2 is bounded. This converges to zero for e.g. kn = n2 . Then the estimator Jn is based on mn = kn n = n3 observations. We define an estimator based on m observations, for every m ∈ N, by J˜m = J m1/3  . A sequence θ˜m = θ + O(m−1/2 ) yields a sequence θn = θ + O(n−3/2 ) on our original scale and hence is covered by the previous calculations. We conclude that, for every θn = θ + O(n−1/2 ),  Eθn ,η (J˜n − ψ˜θn ,η )2 (x) pθn ,η (x) dµ(x) → 0. Thus the sequence J˜n is consistent as desired. To find a sequence of estimators that is both consistent and has small bias, we replace J˜n by, with m = n/2, ˆn,θ (x) = J˜mn (x) + Tn−mn (Xmn +1 , . . . , Xn ) − θ −

1 n − mn

n

J˜mn (Xi ).

i=mn +1

By assumption this is equivalent to J˜mn (x) +

1 n − mn

n

(ψ˜θn ,η − J˜mn )(Xi ) + oP (n−1/2 )

i=mn +1

 J˜mn (x) −

J˜mn pθn ,η dµ + oP (n−1/2 ),

by comparing conditional means and variances given X1 , . . . , Xmn , and where the oP (n−1/2 )-term does not depend on x. Thus the estimator ˆn,θ is both consistent and has small bias.

7.3 Symmetric location Suppose that we observe a random sample from a density η(x − θ) that is symmetric about θ. In Example 2.17 it was seen that the efficient score function for θ is the ordinary score function, 

η ˜θ,η (x) = − (x − θ). η We can apply Theorem 7.2 to construct an asymptotically efficient estimator sequence for θ under the minimal condition that the density η has finite Fisher information for location. First, as an initial estimator θ˜n , we may use a discretized Z-estimator, solving Pn ψ(x − θ) = 0 for a well-behaved, symmetric √ function ψ. For instance, the score function of the logistic density. The n-consistency can be established by the techniques of Lectures 4 and 5.

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419

Second, it suffices to construct estimators ˆn,θ that satisfy (7.2). By symmetry, the variables Ti = |Xi − θ| are, for a fixed θ, sampled from the density g(s) = 2η(s)1{s > 0}. We use these variables to construct an estimator kˆn for the function g  /g, and next we set   ˆn,θ (x; X1 , . . . , Xn ) = −kˆn |x − θ|; T1 , . . . , Tn sign(x − θ). Since this function is skew-symmetric about the point θ, the bias condition in (7.2) is satisfied, with a bias of zero. Since the efficient score function can be written in the form  g  ˜θ,η (x) = − |x − θ| sign(x − θ), g the consistency condition in (7.2) reduces to consistency of kˆn for the function g  /g in that  g  2 P 0. (7.5) (s) g(s) ds → kˆn − g Estimators kˆn can be constructed by several methods, a simple one being the kernel method of density estimation. For a fixed twice continuously differentiable probability density ω with compact support, a bandwidth parameter σn , and further positive tuning parameters αn , βn and γn , set gˆn (s) =

n 1 s − Ti ω , σn i=1 σn

gˆ kˆn (s) = n (s)1Bˆn (s), gˆn   ˆ Bn = s: |ˆ gn (s)| ≤ αn , gˆn (s) ≥ βn , s ≥ γn .

(7.6)

Then (7.2) is satisfied provided αn ↑ ∞, βn ↓ 0, γn ↓ 0 and σn ↓ 0 at appropriate speeds. The proof consists of the usual manipulations of kernel estimators. (See [42], page 398, for a precise statement, or one of the many papers on this model.) This particular construction shows that efficient estimators for θ exist under minimal conditions. It is not necessarily recommended for use in practice. However, any good initial estimator θ˜n and any method of density or curve estimation may be substituted, and will lead to a reasonable estimator for θ, which will be theoretically efficient under some regularity conditions. Open Problem 7.5. It may be verified that the preceding construction generalize to higher dimensions. The problem of estimating θ from a sample of observations from a density η(x − θ) on Rd such that η has finite Fisher information and η(x) = η(−x) is adaptive for any d ≥ 1. Theoretically, one can estimate θ as well knowing η as not knowing η. However, in practice this appears to be nonsense. If d = 10, for instance, it cannot make sense to try and

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7 Lecture: Efficient Score and One-step Estimators

estimate η nonparametrically from n = 1000 observations and the preceding construction will presumably yield bad estimators. The problem is to develop a theory for this phenomenon, maybe using minimax bounds. Note that the problem of estimating θ for d = 10 is by itself not difficult. For instance, we could use an M -estimator and this will be asymptotically normal in the usual way and the asymptotics will be reliable for n ≥ 30. See [32] and [31] for further questions regarding the asymptotic information bounds.

7.4 Errors-in-Variables Let the observations be a random sample of pairs (Xi , Yi ) with the same distribution as X =Z +e Y = α + βZ + f, for a bivariate normal vector (e, f ) with mean zero and covariance matrix Σ and a random variable Z with distribution η, independent of (e, f ). Thus Y is a linear regression on a variable Z which is observed with error. The parameter of interest is θ = (α, β, Σ) and the nuisance parameter is η. To make the parameters identifiable one can put restrictions on either Σ or η. It suffices that η is not normal (where a degenerate distribution is considered normal with variance zero); alternatively it can be assumed that Σ is known up to a scalar. Given (θ, Σ) the statistic ψθ (X, Y ) = (1, β)Σ −1 (X, Y − α)T is sufficient (and complete) for η. This suggests to define estimators for (α, β, Σ) as the solution of the “conditional score equation” Pn ˜θ,ˆη = 0, for

˜θ,η (X, Y ) = ˙θ,η (X, Y ) − Eθ ˙θ,η (X, Y )| ψθ (X, Y ) . This estimating equation has the attractive property of being unbiased in the nuisance parameter, in that Pθ,η ˜θ,η = 0,

every θ, η, η  .

Therefore, the “no-bias” condition is trivially satisfied, and the estimator ηˆ need only be consistent for η (in the sense of (6.7)). One possibility for ηˆ is the maximum likelihood estimator, which was shown to be consistent in Lecture 5 in the case that Σ is known. This proof can be extended to the case that Σ is unknown. As the notation suggests, the function ˜θ,η is equal to the efficient score function for θ. We can prove this by showing that the closed linear span of the set of nuisance scores contains all measurable, square-integrable functions of ψθ (x, y), because then projecting on the nuisance scores is identical to taking the conditional expectation.

7.4 Errors-in-Variables

421

The submodel t → Pθ,tη1 +(1−t)η is well-defined for every 0 ≤ t ≤ 1 and every η1 ∈ H. Its score function is the function pθ,η1 /pθ,η − 1 As is clear from the factorization theorem or direct calculation,  it is a function of the sufficient statistic ψθ (X, Y ). If some function b ψθ (x, y) is orthogonal to all scores of this type and has mean zero, then, for every η1 ,

    pθ,η1 − 1 = 0. Eθ,η1 b ψθ (X, Y ) = Eθ,η b ψθ (X, Y ) pθ,η Consequently, b = 0 almost surely by the completeness of ψθ (X, Y ). We conclude that the closure of the linear span of the nuisance tangent space contains all measurable, square-integrable functions of ψθ (x, y). The efficient score function can be written in the form   ˜θ,η (x, y) = Qθ (x, y) + Pθ (x, y)E Z| ψθ (X, Y ) for polynomials Qθ and Pθ of orders 2 and 1, respectively. The main work is now to show that the class of all functions of this type, when η ranges over a large class of distributions, is Donsker. Because we already know that ηˆ is consistent for the weak topology, it is enough to show this for η ranging over a weak neighbourhood of the true mixing distribution. The following lemma is the main part of the verification. Lemma 7.6. For every 0 < α ≤ 1 and every probability distribution η0 on R and compact K ⊂ (0, ∞), there exists an open neighbourhood U of η0 in the weak topology such that the class F of all functions
2 z ez(b0 +b1 x+b2 y) e−cz dη(z) (x, y) → (a0 + a1 x + a2 y)
z(b +b x+b y) −cz2 , dη(z) e 0 1 2 e with η ranging over U , c ranging over K and a and b ranging over compacta in R3 , satisfies 1 V    5+2α+4/V +δ V /2 log N[] ε, F, L2 (P ) ≤ C P 1 + |x| + |y| , ε for every V ≥ 1/α, every measure P on R2 and δ > 0, and a constant C depending only on α, η0 , U , V , the compacta, and δ. Proof. We only give a sketch of the main steps. See Lemma 7.3 in [25] for the details. First consider the functions
zt2 −cz2 dη(z) ze e . t → gc,η (t) =
zt −cz2 dη(z) e e

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7 Lecture: Efficient Score and One-step Estimators

By some clever applications of Jensen’s and other inequalities it can be proved that there exists a weak neighbourhood U of η0 such that, for η ∈ U and c ∈ K,      gc,η (t) ≤ C(1 + |t|), gc,η (t) ≤ C(1 + |t|)2 . The classical bounds of Kolmogorov give estimates on the covering numbers N (ε, Fj ,  · ) of the classes Fj of all functions f : [j, j + 1] → R such that f ∞ ∨ f  ∞ ≤ Mj for some constant Mj . In the present situation we apply these boundeds with Fj the restrictions of the functions gc,η to the intervals [j, j + 1] and with Mj = (1 + |j|)2 . Given an εj -net fj,1 , . . . , fj,Nj over Fj we can construct brackets for the functions f : R → R by first forming brackets [fj,i − εj , fj,i + εj ] on each interval [j, j + 1) and next glueing these brackets together in every possible combination. Naturally, we choose εj big enough so that for all but finitely many intervals we need to use only one bracket, because otherwise the number of brackets would be infinite. We can optimize the numbers εj and Mj such that resulting brackets on R are ε-brackets relative to the L2 (Q)-norm and such that they are almost a minimal set of ε-brackets, for Q the measure constructed below. For fixed (a, b) the functions fa,b,c,η are essentially the functions gc,η , because fa,b,c,η (x, y) = (a0 + a1 x + a2 y)gc,η (b0 + b1 x + b2 y). A bracket [l, u] for the functions gc,η yields a bracket for the functions fa,b,c,η of the form  (a0 + a1 x + a2 y)+ l(b0 + b1 x + b2 y) − (a0 + a1 x + a2 y)− u(b0 + b1 x + b2 y),  (a0 + a1 x + a2 y)+ u(b0 + b1 x + b2 y) − (a0 + a1 x + a2 y)+ l(b0 + b1 x + b2 y) . Its size in L2 (P ) is equal to the size of [l, u] in L2 (Q) for the measure Q defined by  Q(B) = 1B (b0 + b1 x + b2 y)(a0 + a1 x + a2 y)2 dP (x, y). Thus we can construct the desired brackets for the functions fa,b,c,η as c and η vary, for any fixed value (a, b). For fixed (x, y) the dependence (a, b) → fa,b,c,η (x, y) is Lipschitz of order β = α/2 with Lipschitz constant h(x, y) = (1 + |x| + |y|)2+2β . Now construct brackets over the class of all fa,b,c,η by first choosing an ε1/β /hP,2 -net over the set of all (a, b), next for every (ai , bi ) in this net choose a minimal number of

brackets l, u] over the class of all fai ,bi ,c,η and finally form the brackets l − εh/hP,2 , u + εh/hP,2 ]. Because we need only of the order (1/ε)6/β points (ai , bi ) this last step hardly increases the entropy.

7.4 Errors-in-Variables

423

Notes. Theorem 7.6 is due to [14]. The semiparametric one-step method has a long history, starting with special constructions in the symmetric location model.

8. Lecture: Rates of Convergence

In this lecture we apply maximal inequalities for empirical processes to obtain rates of convergence of minimum contrast estimators, in particular in semiparametric models. These rates are of interest by themselves, but will also be needed to prove the asymptotic normality of semiparametric likelihood estimators in certain models.

8.1 A General Result The set-up is the same as the one in Lecture 5 on consistency. Let Θ be a metric space and for each θ ∈ Θ, let mθ : X → R be a measurable function. Suppose that we are interested in the maximizer θˆ of θ → Pn mθ . We may expect that this converges in probability to the maximizer θ0 of the limiting criterion function θ → P mθ . It is useful to picture the random criterion function Pn mθ as the sum of its limit and the scaled empirical process 1 Pn mθ = P mθ + √ Gn mθ . n Because P mθ is maximal at θ0 we could picture the function θ → P mθ as an inverse parabola with its top at θ0 . Without the second, random term on the right, the estimator θˆ would always choose the top of the parabola, but the fluctuations may pull the maximum of Pn mθ away from θ0 . It is the sizeof the fluctuations  that determines how far. If P mθ ≈ −d(θ, θ0 )2 and ˆ θ0 ) will probably be approxisup Gn mθ : d(θ, θ0 ) ≤ δ ≈ φn (δ), then d(θ, mately equal to the value δ that balances the positive and negative parts of 1 −δ 2 + √ φn (δ). n ˆ θ0 ) ≈ δn for φn (δn ) ≈ √nδ 2 . The following In other words, we expect that d(θ, n theorem makes this precise. As for the consistency results, we do not need θˆ to maximize Pn mθ . We only need that Pn mθˆ ≥ Pn mθ0 .

8.1 A General Result

425

Theorem 8.1. Suppose that, for all sufficiently small d(θ, θ0 ) all sufficiently small δ > 0 and a function φn such that φn (cδ) ≤ cα φn (δ) for all c > 1, and some α < 2, E∗

P (mθ − mθ0 ) ≤ −d2 (θ, θ0 ),   sup Gn (mθ − mθ0 ) ≤ φn (δ).

d(θ,θ0 )<δ

P Then Pn mθˆ ≥ Pn mθ0 and θˆn → imply that d(θˆn , θ0 ) = OP (δn ) √θ0 together 2 for every δn satisfying φn (δn ) ≤ nδn .

Proof. For each n, the parameter space (minus the point θ0 ) can be partitioned into the “shells” Sj,n = {θ: 2j−1 δn < d(θ, θ0 ) ≤ 2j δn } with j ranging over the integers. If d(θˆn , θ0 ) is larger than 2M δn for a given integer M , then θˆn is in one of the shells Sj,n with j ≥ M . In that case the supremum of the map θ → Pn mθ − Pn mθ0 over this shell is nonnegative by the property of θˆn . Conclude that, for every η > 0,



  P∗ d(θˆn , θ0 ) > 2M δn ≤ P∗ sup Pn mθ − Pn mθ0 ≥ 0 j≥M 2j δn ≤η

θ∈Sj,n

  + P∗ 2d(θˆn , θ0 ) ≥ η . Because the sequence θˆn is consistent for θ0 , the second probability on the right converges to 0 as n → ∞ for every η > 0. Choose η > 0 small enough that the first condition of the theorem holds for every d(θ, θ0 ) ≤ η and the second for every δ ≤ η. Then for every j involved in the sum, we have, for every θ ∈ Sj,n , P mθ − P mθ0 ≤ −d2 (θ, θ0 )  −22j−2 δn2 . Therefore, the series may be bounded by j≥M 2j δn ≤η



φn (2j δn )  √ √ 2 2j P∗ Gn (mθ − mθ0 )Sj,n ≥ n22j−2 δn2  nδn 2 j≥M





2jα−2j ,

j≥M

by Markov’s inequality, the definition of δn , and the fact that φn (cδ) ≤ cα φn (δ) for every c > 1. This expression converges to zero for every M = Mn → ∞.

The first condition of the theorem can be expected to hold if θ0 is a point of maximum of θ → P mθ and this function is twice differentiable. More generally, we can see it as simply defining the type of metric that we can

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work with. For instance, if mθ is a log likelihood under parameter θ and P = Pθ0 , then Pθ0 (mθ − mθ0 ) is the Kullback–Leibler divergence and we can either use this directly (inspection of the proof of the theorem, shows that it is not really necessary that d is a metric), or a metric whose square dominates this, such as the Hellinger distance. It is well known that for any pair of probability densities p and q,  √ √ P log(q/p) ≤ −h2 (P, Q) = − p − q)2 dµ. (8.1) Thus the Hellinger distance is a natural distance when considering rates of convergence of maximum likelihood estimators. The latter observation also points out a severe limitation of the theorem: the choice of metrics with which it works is limited. For instance, in a semiparametric model with parameter (θ, η) we might wish to prove that the maximum likelihood estimator, or some other contrast estimator, possesses a √ n-rate of convergence. This will very rarely follow with the help of the preceding theorem, because the theorem will give a rate for the joint estimator ˆ ηˆ), rather than for θˆ only. The joint rate will typically be determined by (θ, √ the rate of ηˆ and this will typically be slower than n. Even a natural rate on the nuisance parameter ηˆ may not be derivable from the theorem, if “natural” refers to a particular, natural distance, which does not combine well with the distance imposed by the theorem. As a consequence, unfortunately, the applicability of the theorem to semiparametric models is limited. The second condition of the theorem requires a maximal inequality for the modulus of the empirical process. Here the inequalities of Lecture 6 may work if the size of the functions mθ − mθ0 is comparable to the size of the envelope function of the class of all such functions with d(θ, θ0 ) < δ. This is not always the case. The following maximal inequalities directly take the size of the functions mθ − mθ0 into account. Lemma 8.2. Let F be a class of measurable functions with f ∞ ≤ M , and P f 2 < δ 2 for every f ∈ F. Then     M J[ ] δ, F, L2 (P ) ∗ √ EP Gn F  J[ ] δ, F, L2 (P ) 1 + . δ2 n The preceding lemma is sufficient for many examples. However, sometimes the assumption that the class is uniformly bounded is restrictive. This can be remedied by computing the size of the brackets relative to a larger norm. Specifically, consider  f P,B = 2P (e|f | − 1 − |f |). The subscript “B” is for Bernstein, as this “norm” is essential in an exponential inequality for sums due to Bernstein, which plays a major role in the

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proofs of maximal inequalities. Actually, the quantity  · P,B is not a norm, as it does not satisfy the triangle inequality. However, we can use  · P,B as a measure of the size of a function and hence as a measure of the size of a bracket [l, u] by applying it to the function u − l. We can define an entropy integral relative to it accordingly. Lemma 8.3. Let F be a class of measurable functions with P f 2 < δ 2 for every f ∈ F. Then     J[ ] δ, F,  · P,B ∗ √ EP Gn F  J[ ] δ, F,  · P,B 1 + . δ2 n

8.2 Nuisance Parameters In this section we consider the same problem of finding an upper bound on ˆ but now in the the rate of convergence of a minimum contrast estimator θ, presence of an estimated nuisance parameter. Using the “wrong”, estimated contrast function should bring the rate of convergence down, but only if the estimation of the nuisance parameter is the harder part of the problem. The following theorem implements this idea. The theorem is of interest not only because it takes care of problems with nuisance parameters of the type as considered before, but also of certain penalized minimum contrast estimators, in which the smoothing parameter of the penalty can be thought of as an estimated nuisance parameter. Consider “estimators” θˆn contained in a metric space Θn satisfying, for a given “estimators” ηˆn contained in a metric space Hn , Pn mθˆn ,ˆηn ≥ Pn mθ0 ,ˆηn for given measurable functions x → mθ,η (x). This is valid, for example, for θˆn equal to the maximizer of the function θ → Pn mθ,ˆηn over Θn , if this set contains θ0 . Assume that the following conditions are satisfied for every θ ∈ Θn , every η ∈ Hn and every δ > 0.   P mθ,η − mθ0 ,η  −d2η (θ, θ0 ) + d2 (η, η0 ), (8.2)   ∗ Gn (mθ,η − mθ ,η )  φn (δ). E sup (8.3) dη (θ,θ0 )<δ,d(η,η0 )<δ θ∈Θn ,η∈Hn

0

Here d2η (θ, θ0 ) may be thought of as the square of a distance, but the following theorem is true for arbitrary functions θ → d2η (θ, θ0 ). Usually dη does not depend on η, but in this form the following theorem is flexible enough to apply to penalized minimum contrast estimators, where the smoothing parameter can be included in η.

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Theorem 8.4. Suppose that (8.3) is valid, for all sufficiently small δ > 0 and a function φn such that φn (cδ) ≤ cα φn (δ) for all c > 1, and some α < 2, ˆ ηˆ) with probability tending to 1. Then and for sets Θn × Hn that contain (θ,  ∗ ˆ θ0 ) = O δn + d(ˆ η , η0 ) for any sequence of positive numbers δn such dηˆ(θ, P√ that φn (δn ) ≤ nδn2 for every n. Proof. For each n ∈ N, j ∈ Z and M > 0 define a set Sn,j,M   = (θ, η) ∈ Θn ×Hn : 2j−1 δn < dη (θ, θ0 ) ≤ 2j δn , d(η, η0 ) ≤ 2−M dη (θ, θ0 ) .  ˆ θ 0 ) ≥ 2M δ n + Then the intersection of the events θˆ ∈ Θn , ηˆ ∈ Hn and dηˆ(θ,    ˆ θ0 ) is contained in the union of the events (θ, ˆ ηˆ) ∈ Sn,j,M over j ≥ d(θ, ˆ the variable sup M . By the definition of θ, Pn (mθ,η − mθ0 ,η ) is  (θ,η)∈Sn,j,M  ˆ ηˆ) ∈ Sn,j,M . Conclude that, for every δ > 0, nonnegative on the event (θ,

  ˆ θ0 ) ≥ 2M δn + d(ˆ P∗ dηˆ(θ, η , η0 ) , θˆ ∈ Θn , ηˆ ∈ Hn

  ≤ P∗ Pn mθ,η − mθ0 ,η ≥ 0 . sup j≥M

(θ,η)∈Sj,n,M

For every j involved in the sum, we have, for every (θ, η) ∈ Sj,n,M and every sufficiently large M ,   P mθ,η − mθ0 ,η  −d2η (θ, θ0 ) + d2 (η, η0 )  −(1 − 2−2M ) d2η (θ, θ0 )  −22j−2 δn2 . We now finish the proof as the proof of Theorem 8.1.

For dη = d not depending on η condition (8.2) is implied by the conditions   P mθ0 ,η − mθ0 ,η0  −d2 (η, η0 ),   P mθ,η − mθ0 ,η0  −d2 (θ, θ0 ). These two conditions are the natural requirement that the criterion function (θ, η) → P mθ,η behaves quadratically (relative to a distance) around the point of maximum (θ0 , η0 ).

8.3 Cox Regression with Current Status Data Let us apply the Theorem 8.1 to one example, which illustrates the potential and difficulties, and for which we shall need the rate of convergence in the next lecture as input to proving asymptotic normality of the maximum likelihood

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estimator. It is again the Cox model, but this time with a type of censoring that changes everything. Suppose that we observe a random sample from the distribution of X = (C, ∆, Z), where ∆ = 1{T ≤ C}, that the “survival time” T and the observation time C are independent given Z, and that T follows a Cox model. The density of X relative to the product of FC,Z and counting measure on {0, 1} is given by  δ  1−δ T T exp(−eθ z Λ(c)) pθ,Λ (x) = 1 − exp(−eθ z Λ(c)) . We define this as the likelihood for one observation x and are interested in the estimator (θˆn , Λˆn ) obtained by maximizing the full likelihood. Here we restrict the parameter θ to a compact Θ ⊂ Rk and restrict the parameter Λ to the set of all cumulative hazard functions with Λ(τ ) ≤ M for a fixed large constant M and τ the “end of the study” (the end point of the distribution of C). We make the following assumptions. The observation time C possesses a Lebesgue density which is continuous and positive on an interval [σ, τ ] and vanishes outside this interval. The true parameter Λ0 is continuously differentiable on this interval, satisfies 0 < Λ0 (σ−) ≤ Λ0 (τ ) < M , and is continuously differentiable on [σ, τ ]. The covariate vector Z is bounded and E cov(Z| C) > 0. The true parameter θ0 is an inner point of the parameter set and the efficient information for θ is positive. (We make the latter condition concrete in the next lecture.) Lemma 8.5. Under the conditions listed previously, θˆn is consistent and  Λˆn − Λ0 P0 ,2 = OP n−1/3 . Actually, we shall show that θˆn also possesses a rate of convergence of at least n−1/3 . However, in the next lecture we shall see that the true rate is n−1/2 . It is a good illustration of what cannot be achieved with the preceding rate theorem. Remembering Trick 1 of Lecture 5 we apply Theorem 8.1 not with mθ equal to the log likelihood (as would be the straightforward thing to do), but with the functions mθ,Λ = log (pθ,Λ + p0 )/2, where the 0 denote the “true” parameter (θ0 , Λ0 ). The densities pθ,Λ are bounded above by 1, and under our assumptions the density p0 is bounded away from zero. It follows that the functions mθ,Λ (x) are uniformly bounded in (θ, Λ) and x, which is of some help. In Lemma 8.6 below we explicitly bound the bracketing numbers of the class of functions mθ,Λ , from which we infer that these are finite. Therefore, the class of functions mθ,Λ forms a Glivenko–Cantelli class. The parameter set Θ is compact by assumption and the parameter set for Λ is compact for the weak topology, also partly because of our assumptions. If the parameter

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(θ0 , Λ0 ) were identifiable, we could conclude by Theorem 5.8 that (θˆn , Λn ) is consistent. However, under our assumptions the parameter is not fully identifiable: the parameter Λ0 is identifiable only on the interval (σ, τ ). We P P ˆ → can still conclude that θˆ → θ0 and that Λ(t) Λ0 (t) for every σ < t < τ . (The ˆ convergence of Λ at the points σ and τ does not appear to be guaranteed.) By (8.1) the Kullback–Leibler divergence P0 (mθ,Λ − m0 ) is dominated by the square Hellinger distance between (pθ,Λ + p0 )/2 and p0 , and this in turn is equivalent to the square Hellinger distance between pθ,Λ and p0 . By a lucky coincidence this distance translates easily in natural distances on θ and Λ. By Lemma 8.7 below, we have P0 (mθ,Λ − m0 )  −θ − θ0 2 − Λ − Λ0 22 . Thus we can take minus the right side as the square distance in Theorem 8.1. We only need to bound the modulus of the empirical process for this distance. By Lemma 8.6 below, the bracketing entropy of the class of functions mθ,Λ is of the order (1/ε). By Lemma 8.2 we can choose the function φn in Theorem 8.1 equal to √ √ δ φn (δ) = δ 1 + 2 √ . δ n This leads to a convergence rate of n−1/3 for both θˆ − θ0  and Λˆ − Λ0 2 . We finish with the technical work in the form of two lemmas. Lemma 8.6. Under the conditions listed previously, there exists a constant C such that, for every ε > 0, 1   log N[] ε, {mθ,Λ , (θ, Λ)}, L2 (P0 ) ≤ C . ε Proof. First consider the class of functions mθ,Λ for a fixed θ. These functions depend on Λ monotonely if considered separately for δ = 0 and δ = 1, Thus a bracket Λ1 ≤ Λ ≤ Λ2 for Λ leads, by substitution, readily to a bracket for mθ,Λ . Furthermore, since this dependence is Lipschitz, there exists a constant D such that  τ  2  2  Λ1 (c) − Λ2 (c) dc. mθ,Λ1 − mθ,Λ2 dFC,Z ≤ D σ

Thus, brackets for Λ of L2 -size ε translate into brackets for mθ,Λ of L2 (Pθ,Λ )size proportional to ε. It is well known that the set of all monotone functions Λ: R → [0, M ] possesses a bracketing entropy of the order 1/ε. Therefore, we can cover the set of all Λ by exp C(1/ε) brackets of size ε. Next, we allow θ to vary freely as well. The partial derivative ∂/∂θ mθ,Λ (x) is uniformly bounded in (θ, Λ, x). Therefore, if mθ,Λ is contained in a bracket [l, u], then mθ ,Λ is contained in the bracket [l − ε, u + ε] for every θ with

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θ − θ  ε. If the bracket [l, u] is of size ε, then the bracket [l − ε, u + ε] is of size 2ε. It follows that we can construct a set of brackets for the functions mθ,Λ by first selecting an ε-net θ1 , . . . , θp over Θ, then apply the procedure of the first paragraph to find brackets for the functions mθi ,Λ for each i, and finally enlarging this bracket. The total number of brackets will be of the

order (1/ε)k exp c(1/ε). Lemma 8.7. Under the conditions listed previously there exist constants C, ε > 0 such that, for all Λ and all θ − θ0  < ε,  τ   1/2 1/2 2 (Λ − Λ0 )2 (c) dc + Cθ − θ0 2 . pθ,Λ − pθ0 ,Λ0 dµ ≥ C σ

Proof. The left side of the lemma can be rewritten as  (pθ,Λ − pθ0 ,Λ0 )2 dµ.  1/2 1/2 2 pθ,Λ + pθ0 ,Λ0 Since p0 is bounded away from zero, and the densities pθ,Λ are uniformly bounded, the denominator can be bounded above and below by positive constants. Thus the Hellinger distance (in the display) is equivalent to the L2 distance between the densities, which can be rewritten   2 θT z θT z 2 e−e Λ(c) − e−e 0 Λ0 (c) dF Y,Z (c, z). T

Let g(t) be the function exp(−eθ z Λ(c)) evaluated at θt = tθ + (1 − t)θ0 and Λt = tΛ  + (1 − t)Λ0 , for fixed (c, z). Then the integrand is equal to g(1) − g(0) 2 , and hence, by the mean value theorem, there exists 0 ≤ t = t(c, z) ≤ 1 such that the preceding display is equal to   2 T   −Λt (c)eθt z θtT z T T (Λ − Λ0 )(c) 1 + t(θ − θ0 ) z + (θ − θ0 ) zΛ0 (c) P0 e e . θT z

Here the multiplicative factor e−Λt (c)e t eθt z is bounded away from zero. By dropping this term we obtain, up to a constant, a lower bound for the left side of the lemma. The remainder of the proof is best understood in terms of semiparametric information. We adopt the notation of the information calculations given in the next lecture. Since the function Qθ0 ,Λ0 is bounded away from zero and infinity, we may add a factor Q2θ0 ,Λ0 , and obtain the lower bound, up to a constant,

2   P0 1 + t(θ − θ0 )T z Bθ0 ,Λ0 (Λ − Λ0 )(x) + (θ − θ0 )T ˙θ0 ,Λ0 (x) . T

Here Bθ0 ,Λ0 is the score operator for the model, which we derive in the next   lecture. The function h = 1 + t(θ − θ0 )T z is uniformly close to 1 if θ is close to θ0 . Furthermore, for any function g and vector a,

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P0 (Bθ0 ,Λ0 g)aT ˙θ0 ,Λ0

2

 2 = P0 (Bθ0 ,Λ0 g)aT (˙θ0 ,Λ0 − ˜0 )  2 ≤ P0 Bθ0 ,Λ0 g aT (I0 − I˜0 )a,

by the Cauchy–Schwarz inequality. Since the efficient information I˜0 is positive-definite by assumption, the term aT (I0 − I˜0 )a on the right can be written aT I0 ac for a constant 0 < c < 1. The lemma now follows by application of Lemma 8.8 ahead. Lemma 8.8. Let h, g1 and g2 be measurable functions such that c1 ≤ h ≤ c2 and (P g1 g2 )2 ≤ cP g12 P g22 for a constant c < 1 and constants c1 < 1 < c2 close to 1. Then P (hg1 + g2 )2 ≥ C(P g12 + P g22 ), √ for a constant C depending on c, c1 and c2 that approaches 1 − c as c1 ↑ 1 and c2 ↓ 1. Proof. We may first use the ineqalities 2 (hg1 + g2 )2 ≥ c1 hg12 + 2hg1 g2 + c−1 2 hg2 2 = h(g1 + g2 )2 + (c1 − 1)hg12 + (1 − c−1 2 )hg2 2 ≥ c1 (g12 + 2g1 g2 + g22 ) + (c1 − 1)c2 g12 + (c−1 2 − 1)g2 .

Next, we integrate this with respect to P , and use the inequality for P g1 g2 on the second term to see that the left side of the lemma is bounded below by  2 c1 (P g12 − 2 cP g12 P g22 + P g22 ) + (c1 − 1)c2 P g12 + (c−1 2 − 1)c2 P g2 . Finally, we apply the inequality 2xy ≤ x2 + y 2 on the second term.



Notes. Rates of convergence have been a hot topic in the 1990s. Here we have only said enough in order to be able to treat the Cox model with current status censoring in Lecture 9. The papers [5] and [6] are important contributions and contain good references. Another source of references is the book [41], which also gives an overview.

9. Lecture: Maximum and Profile Likelihood

In this lecture we study likelihood methods for semiparametric models. This concerns both ordinary likelihoods indexed by infinite-dimensional parameters and empirical likelihoods.

9.1 Examples “Likelihood” is the key unifying element in classical statistics and hence it is worth while to seek a theory of likelihood for semiparametric models. This will be the subject of our last two lectures. Unfortunately, what we shall have to say is not completely satisfying. As known today likelihood theory for semiparametric models falls short of the beautiful and simple theory for parametric models. A first problem is that it is not obvious what we should define to be the “likelihood” of a given semiparametric model, in general. It is obvious that the likelihood has something to do with a density of the observations, viewed as function of the parameter. Apart from the fact that we also need to choose particular versions of these densities, we encounter the further, major problem that many semiparametric models are not dominated, or are defined in terms of densities that maximize to infinity. The good news is that given a concrete example it is usually not difficult to choose a “likelihood”, albeit that often other, slightly different choices would be just as reasonable. Sometimes a likelihood can be taken equal to a density with respect to a dominating measure, for other models we use an “empirical likelihood”, but mixtures of these situations occur as well, and sometimes it is fruitful to incorporate a “penalty” in the likelihood, yielding a “penalized likelihood estimator”, maximize the likelihood over a set of parameters that changes with n, yielding a “sieved likelihood estimator”, or group the data in some way before writing down a likelihood. To bring out this difference with the “classical”, parametric maximum likelihood estimators, some authors use the phrase “nonparametric maximum likelihood estimators” (NPMLE). We prefer to speak simply of “maximum likelihood estimators”, accepting the risk of being charged that nothing new is happening here. (In fact, it would be nice if nothing new needed to happen.) After all, in each of the models we are thinking of there is only one likelihood. We shall not

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give an abstract definition of “likelihood”, but shall describe “likelihoods that work” for a number of examples to set the stage. We denote the likelihood for the parameter P given one observation x by lik(P )(x) or lik(θ, η) if P = Pθ,η . Given a measure P , write P {x} for the measure of the one-point set {x}. The function x → P {x} may be considered the density of P , or its absolutely continuous part, with respect to counting measure. The empirical likelihood of a sample X1 , . . . , Xn is the function, P →

n 

P {Xi }.

i=1

Given a model P, a maximum likelihood estimator could be defined as the distribution Pˆ that maximizes the empirical likelihood over P. Such an estimator may or may not exist. Example 9.1 (Empirical distribution). Let P be the set of all probability distributions on the measurable space (X , A) (in which one-point sets . , xn , the vector are for n fixed different values x1 , . .   measurable). Then, vectors p ≥ 0 such that pi ≤ 1 when P P {x1 }, . . . , P {xn } ranges over all  p , it is clearly best to choose p maximal: ranges over P. To maximize p → i i  p = 1. Then, by symmetry, the maximizer must be p = (1/n, . . . , 1/n). i i  Thus, the empirical distribution Pn = n−1 δXi maximizes the empirical likelihood over the nonparametric model, whence it is referred to as the nonparametric maximum likelihood estimator. If there are ties in the observations, this argument must be adapted, but the result is the same. The empirical likelihood is appropriate for the nonparametric model. For instance, in the case of a Euclidean space, even if the model would be restricted to distributions with na continuous Lebesgue density p, then we still could not use the map p → i=1 p(Xi ) as a likelihood. The supremum of this “likelihood” is infinite, for we could choose p to have an arbitrarily high, very thin peak at some observation. n Open Problem 9.2. Suppose we use p → i=1 p(Xi ) as a likelihood, restricted to a H¨older ball of densities p: [0, 1] → R, e.g. all densities which are twice continuously differentiable with second derivative bounded by 1 and
which are themselves bounded by some fixed number. Is it true that h(x) pˆ(x) dx is an asymptotically efficient estimator for ψ(P ) =
h(x) p(x) dx for every reasonable function h? Example 9.3 (Cox model). We already discussed the problem of finding a likelihood for the Cox model in Lecture 5. There we settled on using the function

δ θz

1−δ θz lik(θ, Λ)(y, δ, x) = eθz Λ{y}e−e Λ(y) . e−e Λ(y)

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We also agreed to maximize this over all θ and over all nondecreasing, cadlag functions Λ with Λ(0) = 0. This is close, but not quite an empirical likelihood. Furthermore, we have enlarged the parameter set slightly, by not restricting the jumps of Λ to be at most 1. At the end of this lecture, when discussing profile likelihood, we reveal the reason for the latter. Example 9.4 (Mixtures). Mixture models usually are based on well-behaved parametric families of densities, and then lead to well-behaved likelihoods equal to the ordinary density. Thus for a given kernel pθ (·| z) and pθ,η the corresponding mixture density we simply set lik(θ, η)(x) = pθ,η (x). Surprisingly little is known about the behaviour of such likelihoods. For example, it is known for only a handful of examples that the θ-component of the ˆ ηˆ) is asymptotically efficient for estimating maximum likelihood estimator (θ, θ, as we would certainly expect. Open Problem 9.5. Just to show how little is known. Suppose that
X1 , . . . , Xn are sampled from a normal location mixture pη (x)
= φ(x−z) dz and let ηˆ be the maximum likelihood estimator for η. Then z dˆ η (z) = X n (as can be ascertained by manipulation of likelihood equations) and hence
z dˆ η (z) is asymptotically efficient for estimating the mean of η, if this exists.
η (z)? Is the analogous statement true for the higher moments z k dˆ Example 9.6 (Penalized logistic regression). In this model we observe a random sample from the distribution of X = (V, W, Y ), for a 0-1 variable Y that follows the logistic regression model   Pθ,η (Y = 1| V, W ) = Ψ θV + η(W ) , where Ψ (u) = 1/(1 + e−u ) is the logistic distribution function. Thus, the usual linear regression of (V, W ) has been replaced by the partial linear regression θV + η(W ), where η ranges over a large set of “smooth functions”. For instance, η is restricted to the Sobolev class of functions on [0, 1] whose (k − 1)st derivative exists and is absolutely continuous with J(η) < ∞, where  1  (k) 2 2 η (w) dw. J (η) = 0

Here k ≥ 1 is a fixed integer and η (k) is the kth derivative of η with respect to z. The density of an observation is given by 1−y  y  fV,W (v, w). pθ,η (x) = Ψ θv + η(w) 1 − Ψ (θv + η(w) We cannot use this directly for defining a likelihood. The resulting maximizer ηˆ would be such that ηˆ(wi ) = ∞ for every wi with yi = 1 and ηˆ(wi ) = −∞

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when yi = 0, or, at least we could construct a sequence of finite, smooth ηm approaching this extreme choice. The problem is that qualitative smoothness assumptions such as J(η) < ∞ do not restrict η on a finite set of points w1 , . . . , wn in any way. To remedy this situation we could restrict the maximization to a smaller set of η, which we could allow to grow as n → ∞. For instance, the set of all η such that J(η) ≤ Mn for Mn ↑ ∞ at a slow rate, or a sequence of spline approximations. An alternative is to use a penalized likelihood, of the form ˆ 2 J 2 (η). (θ, η) → Pn log pθ,η − λ n ˆ n is a “smoothing parameter” that determines the importance of the Here λ ˆ n will lead to smooth maximizers ηˆ, while penalty J 2 (η). A large value of λ for small values the maximizer will be more like the unrestricted maximum likelihood estimator. Intermediate values are best, and are often chosen by a data-dependent scheme, such as cross validation.

9.2 Asymptotic Normality There are two ways of proving of asymptotic normality of the maximum likelihood estimator in parametric models: one based on maximization and one based on the likelihood equations. We like the first proof better, but it appears to be hard to generalize it to general semiparametric models, with its different types of likelihoods and possibly hard to estimate nuisance parameters. The proof based on the likelihood equations is easier to adapt to semiparametric models. If we are interested in the behaviour of the maximum likelihood estimator for θ in a semiparametric model with parameter (θ, η), then we have two possibilities. The first is to set up a system of likelihood equations for both parameters θ and η and infer the joint asymptotic normality of the maximum likelihood estimators. We shall discuss this method in the last lecture, Lecture 10. The second possibility is to treat η as a nuisance parameter in the likelihood equation for θ. In fact, if θˆ would satisfy the efficient score equation discussed in Lecture 7, then we have already proved its asymptotic normality and efficiency, under some conditions. Sometimes the analysis is this easy, but not in general. Perhaps unexpectedly, the efficient score function may not be a “proper” score function and the maximum likelihood estimator may not satisfy the efficient score equation. This is becasue, by definition, the efficient score function is a projection (and L2 -approximation), and nothing guarantees that this projection is the derivative of the log likelihood along some submodel. If there exists a “least ˆ ηˆ) such that η0 (θ, ˆ ηˆ) = ηˆ, and, for every x, favourable” path t → ηt (θ,

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  ∂ ˆ ηˆ) (x), log lik θˆ + t, ηt (θ, ˜θ,ˆ ˆ η (x) = ∂t |t=0 then the maximum likelihood estimator satisfies the efficient score equation; if not, then this is not clear. The existence of an exact least favourable submodel appears to be particularly uncertain at the maximum likelihood estimator ˆ ηˆ), as this tends to be on the “boundary” of the parameter set. (θ, A method around this difficulty is to replace the efficient score equation ˆ ηˆ) satisfies the efficient score by an approximation. First, it suffices that (θ, equation approximately, for Theorem 7.1 goes through for every consistent √ = o estimator sequence θˆ such that nPn ˜θ,ˆ ˆη P (1). Second, this theorem is based on the more general Theorem 6.20, which yields asymptotic normality of estimators satisfying a more general estimating equation Pn ψθ,ˆη ≈ 0, and actually uses the special property of an efficient score function only to reduce the asymptotic variance to the inverse efficient influence function. As long as we can show that the maximum likelihood estimator θˆ satisfies an equation Pn ψθ,ˆ ˆ η ≈ 0 for functions ψθ,η that, if evaluated at the true parameter (θ, η), give the efficient score function, then we still can conclude that θˆ is asymptotically efficient. This motivates us to introduce approximately least favourable subprovided models. Definition 9.7. An approximately least favourable subprovided models is a collection of maps t → ηt (θ, η) from a neighbourhood of 0 ∈ Rk to the parameter set for η with η0 (θ, η) = η (for every (θ, η)) such that ψθ,η (x) =

  ∂ log lik θ + t, ηt (θ, η) (x), ∂t |t=0

exists (for every x) and is equal to the efficient score function at (θ, η) = (θ0 , η0 ). Thus, the path t → ηt (θ, η) must pass through η at t = 0, and at the true parameter (θ0 , η0 ) the submodel is truly least favourable in that its score is the efficient score for θ. We need such a submodel for every fixed (θ, η), or at ˆ ηˆ). least for the true value (θ0 , η0 ) and every possible value of (θ, ˆ ηˆ) maximizes the likelihood, then the function If (θ,   ˆ ηˆ) t → Pn log lik θ + t, ηt (θ, ˆ ηˆ) satisfies the stationary equation is maximal at t = 0 and hence (θ, ˆ Pn ψθ,ˆ ˆ η = 0. Now Theorem 6.20 yields the asymptotic efficiency of θn . The main assumptions are that the entropies of the classes of realizations of the functions ψθ,ˆ ˆ η are stable and not too big, and the no-bias and consistency conditions (6.6) and (6.7). Two obvious questions arise:

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– Does an approximately least favourable submodel always exist? – If it exists can it be chosen to satisfy the “regularity” conditions, such as (6.6)? We discussed the nature of (6.6) in Lecture 6 and have nothing to add to it. We do not have a satisfying answer to the first question either. In many examples such submodels exist, but we have already mentioned some examples where the question of asymptotic normality of the maximum likelihood estimator is still open. To give some insight in the difficulties we discuss one example in some detail below. More in general, we note that we can often use our insight in the calculus of scores developed in the preceding lectures. Assume, for instance, that the information operator B0∗ B0 , evaluated at the true parameter (θ0 , Λ0 ) is continuously invertible. Then the efficient score function is given by ˜0 = ˙0 − B0 (B0∗ B0 )−1 B0∗ ˙0 . A score function Bθ,η h would presumably arise from some path t → ηt (η)(h) in the H-space. Then a potential least favourable path is given by ηt (θ, η) = ηt (η)(−h0 ),

h0 = (B0∗ B0 )−1 B0∗ ˙0 .

This, of course, is only possible if h0 is a valid direction for perturbation of η in the H-space. It may be necessary to recenter h0 first, and we may have to ascertain that h0 is a nice function, e.g. bounded, or continuous, to make the path well-defined. Example 9.8 (Cox model). A convenient approximately least favourable submodel in the Cox model is defined by   dΛt (θ, Λ) = 1 − th0 dΛ, where h0 = L1,θ0 /L0,θ0 is the least favourable direction in the Λ-space at the true parameter (θ0 , Λ0 ). (See Example 3.10.) This is a valid cumulative hazard function, at least for t ≈ 0, if h0 is a bounded function, and this is true for instance if Z ranges over a bounded interval. Substituting this submodel in Cox likelihood and differentiating with respect to t gives ψθ,Λ (x) =

  ∂ lik θ + t, Λt (θ, λ) (x) = ˙θ,Λ − Bθ,Λ h0 (x). ∂t t=0

This is not the efficient score function at every choice (θ, Λ), but it is the efficient score function for (θ, Λ) = (θ0 , Λ0 ), which is enough. The regularity conditions of Theorem 6.20 can be verified. Let us restrict ourselves to the most interesting one, the no-bias condition (6.8). We have Pθ0 ,Λ0 ψθ0 ,Λˆ = Pθ0 ,Λ0 (˙θ0 ,Λˆ − Bθ0 ,Λˆh0 )

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 = Pθ0 ,Λ0 (˙θ0 ,Λˆ − Bθ0 ,Λˆh0 ) − (˙θ0 ,Λ0 − Bθ0 ,Λ0 h0 ) 

 = −Pθ0 ,Λ0 zeθ0 z (Λˆ − Λ0 )(y) − eθ0 z h0 d(Λˆ − Λ0 ) [0,y]  = − (L1,θ0 − L0,θ0 h0 ) d(Λˆ − Λ0 ). The right side vanishes by the definition of the least favourable direction h0 . Therefore, the “no bias” condition is satisfied in the strongest possible sense. (We could have inferred this immediately from the linearity of the score functions in Λ (even though the likelihood is not linear in Λ)). Again, the Cox model is as nice as it can be; in other cases we do find a remainder term, and need to establish some rate of convergence.

9.3 Cox Regression with Current Status Data We take up the example for which we computed rates of convergence in Lecture 8. Thus we observe a random sample from the density  δ  1−δ T T pθ,Λ (x) = 1 − exp(−eθ z Λ(c)) exp(−eθ z Λ(c)) . We define this density as the likelihood for one observation x = (c, δ, z). We make the same assumptions as in Lecture 8, but add the assumption that the function hθ0 ,Λ0 given by (9.1) ahead has a version which is differentiable with a bounded derivative on [σ, τ ]. The score function for θ takes the form ˙θ,Λ (x) = zΛ(c)Qθ,Λ (x), for the function Qθ,Λ given by θT z

Qθ,Λ (x) = e

 δ

e−e

θT z

1 − e−e

Λ(c)

θT z

Λ(c)

 − (1 − δ) .

For every nondecreasing, nonnegative function h and positive number t, the submodel Λt = Λ + th is well defined. Inserting this in the log likelihood and differentiating with respect to t at t = 0, we obtain a score function for Λ of the form Bθ,Λ h(x) = h(c)Qθ,Λ (x). The linear span of these score functions contains Bθ,Λ h for all bounded functions h of bounded variation. In view of the similar structure of the scores for θ and Λ, projecting ˙θ,Λ onto the closed linear span of the nuisance scores is a weighted least squares problem with weight function Qθ,Λ . The solution is given by the vector-valued function

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  Eθ,Λ ZQ2θ,Λ (X)| C = c   . hθ,Λ (c) = Λ(c) Eθ,Λ Q2θ,Λ (X)| C = c

(9.1)

The efficient score function for θ takes the form   ˜θ,Λ (x) = zΛ(c) − hθ,Λ (c) Qθ,Λ (x). Formally, this function is the derivative at t = 0 of the log likelihood evaluated at (θ + t, Λ − tT hθ,Λ ). However, the second coordinate of the latter path may not define a nondecreasing, nonnegative function for every t in a neighbourhood of 0 and hence cannot be used to obtain a stationary equation for the maximum likelihood estimator. This is true in particular, for discrete cumulative hazard functions Λ, for which Λ + th is nondecreasing for both t < 0 and t > 0 only if h vanishes between the jumps of Λ. This suggests that the maximum likelihood estimator does not satisfy the ˆ we replace efficient score equation. To prove the asymptotic normality of θ, this equation by an approximation, obtained from an approximately least favourable submodel. Our second guess on a least favourable submodel is to use Λt (θ, Λ) = Λ − thθ0 ,Λ0 ◦ Λ−1 0 ◦ Λ. This alleviates the problem of different supports of Λ is Lipschitz, then and its perturbation. Indeed, if the function hθ0 ,Λ0 ◦ Λ−1 0 for any a ≤ b and C the Lipschitz constant,   Λt (θ, Λ)(a) − Λt (θ, Λ)(b) ≤ Λ(a) − Λ(b) (1 − tC). Hence the function Λt (θ, Λ) is nondecreasing for sufficiently small |t|. However, it is not clear that the range of Λt (θ, Λ) is inside [0, M ], whereas we have decided to maximize only over functions with range inside this interval. (It would have been better at this point to drop that restriction, to maximize over all nondecreasing functions, and next to prove that the maximizers remain uniformly bounded with high probability. However, we imposed the restriction to [0, M ] precisely, because we do not know if the last is true. Now we have to pay for it.) This motivates a third guess of a least favourable submodel. We take it to be, with φ a suitably chosen function, Λt (θ, Λ) = Λ − tφ ◦ Λhθ0 ,Λ0 ◦ Λ−1 0 ◦ Λ. If φ is Lipschitz, then Λt (θ, Λ) is nondecreasing, by the same argument as before. If y − tφ(y)hθ0 ,Λ0 ◦ Λ−1 0 (y) is contained in [0, M ] for all y in the range Λ) takes its values in [0, M ]. We achieve this if 0 ≤ φ(y) ≤ of Λ, then Λt (θ,  c y ∧ (M − y) for every 0 ≤ y ≤ M , Under our assumptions we can choose φ in such a way that this holds and, moreover, φ is the identity on the range [Λ0 (s), Λ(τ )] of Λ0 (which is strictly contained in [0, M ]). Inserting θ + t, Λt (θ, Λ) into the log likelihood, and differentiating with respect to t at t = 0, yields the score function     ) Λ(c) ψθ,Λ (x) = zΛ(c) − φ Λ(c) (hθ0 ,Λ0 ◦ Λ−1 Qθ,Λ (x). 0

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When evaluated at (θ0 , Λ0 ) this reduces to the efficient score function ˜θ0 ,Λ0 (x) provided φ(Λ0 ) = 1, whence the submodel is approximately least favourable. To prove the asymptotic efficiency of θˆn it suffices to verify the conditions of Theorem 6.20. To verify the no-bias condition (6.8) we can use the decomposition (6.5) in combination with the inequalities

 pθ

0 ,Λ

− pθ0 ,Λ0

|pθ0 ,Λ − pθ0 ,Λ0 |(x)  |Λ − Λ0 |(c),  ψθ ,Λ − ψθ ,Λ |(x)  |Λ − Λ0 |(c), 0 0 0  − Bθ0 ,Λ0 (Λ − Λ0 )pθ0 ,Λ0 (x)  |Λ − Λ0 |2 (c).

For every fixed x, the expressions on the left depend on Λ only through the scalar Λ(y). For this reason these inequalities follow from ordinary Taylor expansions and uniform bounds on the first and second derivatives. By writing the bias as in (6.5), we now easily obtain that  τ |Pθ0 ,Λ0 ψθ0 ,Λˆ|  |Λˆ − Λ0 |2 (c) dc. σ

The right side was shown to be of the order OP (n−2/3 ) in Lecture 8, and this ˆ is better than the oP (n−1/2 ) that is needed for asymptotic efficiency of θ. The functions ψθ,Λ can be written in the form ψθ,Λ (x) = ψ(z, eθ

T

z

, Λ(c), δ),

for a function ψ that is Lipschitz in its first three coordinates, for δ ∈ {0, 1} fixed. (Note that Λ → ΛQθ,Λ is Lipschitz, as Λ → hθ0 ,Λ0 ◦ Λ−1 0 (Λ)/Λ = T (Λ).) The functions z → z, z → exp θ z, c → Λ(c) and (hθ0 ,Λ0 /Λ0 ) ◦ Λ−1 0 δ → δ form Donsker classes when θ and Λ range freely. Hence the functions x → Λ(c)Qθ,Λ (x) form a Donsker class, by Theorem 6.10. Open Problem 9.9. Find the limit distribution (if any) of the sequence n1/3 (Λˆ − Λ)(t).

9.4 Profile Likelihood Given a partitioned parameter (θ, η) and a likelihood lik(θ, η)(x) the profile likelihood for θ is defined as the function θ → proflik(θ): = sup η

n 

lik(θ, η)(Xi ).

i=1

The supremum is taken over all possible values of η, or given a sieve all values in the sieve at “time” n. It is rarely possible to compute a profile likelihood explicitly, but its numerical evaluation is often feasible.

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The profile likelihood can be used as a computational device, because its point of maximum is exactly the first coordinate of the maximum likelihood ˆ ηˆ). We are simply computing the maximum of the likelihood estimator (θ, over (θ, η) in two steps. However, the importance of the profile likelihood goes far beyond computational issues. Profile likelihood functions can be used in the same way as (ordinary) likelihood functions of parametric models. Besides defining the ˆ the curvature of the log profile likelihood maximum likelihood estimator θ, ˆ at θ can be used as an estimate of minus the inverse of the asymptotic covariˆ Furthermore, the quotient proflik(θ)/ ˆ proflik(θ0 ) between ance matrix of θ. the maximum value and the value at a fixed point θ0 is the likelihood ratio statistic for testing the (composite) null hypothesis H0 : θ = θ0 . In this section we study these quantities more closely. It is well known that for parametric models with log likelihood θ (x) = log lik(θ) the likelihood ratio statistic 2nPn (θˆ − θ0 ) is under some assumptions and under the null hypothesis H0 : θ = θ0 asymptotically chisquared distributed with degrees of freedom equal to the dimension of the parameter. Furthermore, it is well known that the observed information −Pn ¨θˆ is, under some conditions, a consistent estimator of the Fisher information Iθ = Pθ ˙θ ˙Tθ = −Pθ ¨θ . Under some (more) conditions we can prove analogous results for semiparametric models, but with the profile likelihood function for θ replacing the ordinary likelihood. At the basis of these results is an asymptotic expansion of the (profile) P likelihood function as follows. For any random sequence θ˜n → θ0 , log proflikn (θ˜n ) = log proflikn (θ0 ) + (θ˜n − θ0 )T

n

˜θ0 ,η0 (Xi )

i=1

− 12 n(θ˜n − θ0 )T I˜θ0 ,η0 (θ˜n − θ0 ) + oPθ0 ,η0

√

2 nθ˜n − θ0  + 1 . (9.2)

If the maximum likelihood estimator is asymptotically efficient, then it possesses the asymptotic expansion √

1 ˜−1 ˜ I n(θˆn − θ0 ) = √ θ ,η (Xi ) + oPθ0 ,η0 (1). n i=1 θ0 ,η0 0 0 n

(9.3)

Taking this into account we see that the parabolic approximation to the log profile likelihood given by equation (9.2) is centered, to the first order, at θˆn . In other words, it is possible to expand the log profile likelihood function around θˆn , in the form log proflikn (θ˜n ) = log proflikn (θˆn ) − 12 n(θ˜n − θˆn )T I˜θ0 ,η0 (θ˜n − θˆn ) √ 2 + oPθ0 ,η0 nθ˜n − θ0  + 1 .

(9.4)

Actually (9.3)–(9.4) are a consequence of (9.2), as we prove below. The expansion (9.2) is firmly believed to be true in some generality. We shall not

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give precise conditions for its validity here, but note that such conditions have been given in terms of the existence of approximately least favourable paths, much in the spirit of our treatment of maximum likelihood estimators earlier in this lecture. The asymptotic expansions (9.2) and (9.4) justify using a semiparametric profile likelihood as an ordinary likelihood, at least asymptotically. In particular, we present three corollaries. We assume that the true parameter θ0 is interior to the parameter set. Corollary 9.10. If (9.2) holds, I˜θ0 ,η0 is invertible, and θˆn is consistent, then (9.3)–(9.4) hold. In particular, the maximum likelihood estimator θˆ is asymptotically efficient at (θ0 , η0 ). Corollary 9.11. If (9.2) holds, I˜θ0 ,η0 is invertible, and θˆn is consistent, then  ˆ under the null  hypothesis H0 : θ = θ0 , then the sequence 2 log proflikn (θn )/ proflikn (θ0 ) is asymptotically chi-squared distributed with d degrees of freedom. Corollary 9.12. If (9.2) holds and θˆ√ n is consistent, then, for all sequences P P v ∈ Rd and hn → 0 such that ( nhn )−1 = OP (1), vn → −2

log proflikn (θˆn + hn vn ) − log proflikn (θˆn ) P T ˜ → v I0 v. nh2n

Proofs. The second and third corollaries are immediate consequences of (9.2)–(9.4). Relation (9.4) follows from (9.2)–(9.3) and some algebra. We shall n ˆ = √n(θˆ − θ0 ). derive (9.3) from (9.2). Set ∆n = n−1/2 i=1 ˜θ0 ,η0 (Xi ) and h ∆n , we Applying (9.2) with the choices θ˜ = θˆ and θ˜ = θ0 + n−1/2 I˜θ−1 0 ,η0 find ˆ log proflikn (θ)

  ˆ + 1 2, ˆ T ∆n − 1 h ˆ T I˜θ ,η h ˆ + oP h = log proflikn (θ0 ) + h 0 0 2 log proflikn (θ0 + n−1/2 I˜−1 ∆n ) = log proflikn (θ0 ) +

θ0 ,η0 ∆Tn I˜θ−1 ∆n 0 ,η0

− 12 ∆Tn I˜θ−1 ∆n + oP (1). 0 ,η0

ˆ the expression on the left (and hence on the right) By the definition of θ, in the first equation is larger than the expression on the left in the second equation. It follows that   ˆ + 1 2. ˆ T ∆n − 1 h ˆ T I˜θ ,η h ˆ − 1 ∆T I˜−1 ∆n ≥ −oP h h 0 0 2 2 n θ0 ,η0 The left side of this inequality is equal to ˆ − I˜−1 ∆n )T I˜0 (h ˆ − I˜−1 ∆n ) ≤ −ch ˆ − I˜−1 ∆n 2 , − 12 (h θ0 ,η0 θ0 ,η0 θ0 ,η0

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for a positive constant c, by the nonsingularity of I˜θ0 ,η0 . Conclude that   ˆ − I˜−1 ∆n  = oP h ˆ +1 . h θ0 ,η0 ˆ = OP (1), and next, by reinsertion, that h ˆ− This implies first that h −1 ˜

Iθ0 ,η0 ∆n  = oP (1). This completes the proof of (9.3). Example 9.13 (Cox model). Consider again the Cox model of Example 3.10. In Lecture 5 we noted that the second component of the maximum likelihood ˆ Λ), ˆ relative to the likelihood chosen there, will be a step function estimator (θ, with steps only at the Yi such that ∆i = 0. It follows that the profile likelihood function takes the form θ→

sup

n 

λ1 ,...,λn ≥0 i=1

eθZi λi

∆ i

−eθZi


j:Yj ≤Yi

e

λj ∆j

.

In this (very special) case the supremum can be explicitly computed. Finding the maximizers over (λ1 , . . . , λn ) is equivalent to maximizing n

∆i log λi −

n

eθZi λj ∆j .

i=1 j:Yj ≤Yi

i=1

Interchanging the sums and next taking the partial derivative relative to λk for a k such that ∆k = 1, yields the stationary equation 1 = eθZi . λk i:Yi ≥Yk

Upon inserting this in the likelihood we find the profile likelihood for θ θ→

n  i=1



∆ i

eθZi

j:Yj ≥Yi

eθZj


n

e−

i=1 ∆i

.

This expression is known as the Cox partial likelihood. Cox’s original motivation for this criterion function is that the terms in the product are the conditional probabilities that the ith subject dies at time Yi given that one of the subjects at risk dies at that time. The Cox partial log likelihood is a sum over the observations, but the terms in the sum are dependent. Direct study of such a sum therefore is nontrivial at first sight. Initially the Cox partial likelihood estimator was studied along the classical lines: characterizing θˆ as the solution of the derivative of the partial likelihood and next using Taylor series arguments on this partial score equation. The difficulty is then to show that the partial score and its derivative are asymptotically normal or satisfy a law of large numbers. Later it turned out that martingale arguments can both justify this derivation and

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facilitate the calculation of means and variances. Elegant as this arguments are, they are restricted to a special case such as the Cox model. In the final lecture we shall show how the asymptotic normality of the Cox estimators can be derived within a framework that applies to general semiparametric models. Alternatively, the asymptotic normality of θˆ follows along the lines of the present lecture. Notes. The treatment of the Cox model with current status data follows [12], who also presents a general set-up. Our definition of approximately least favourable submodels is based on [40] and [26]. The latter paper discusses the profile likelihood function and summarizes other work on the likelihood ratio statistic and the observed information. For an analysis of the sieved or penalized logistic regression model see [13] and [21].

10. Lecture: Infinite-dimensional Z-Estimators

In this lecture we consider infinite-dimensional systems of estimating equations and show that solutions are asymptotically normal if the system is appropriately differentiable, extending the results on finite-dimensional Zestimators to infinite dimensions. Next we show that this method can be applied to proving asymptotic normality of maximum likelihood estimators in semiparametric models, with as example, again, the Cox model.

10.1 General Result A system of estimating equations for a parameter must be of the same dimension as the parameter. For an infinite-dimensional parameter we need infinitely many estimating equations. It turns out that such a system can be analyzed much in the same way as a finite-dimensional system, provided that we substitute functional analysis for multivariate calculus. The system is linearized in the estimators by a Taylor expansion around the true parameter, and the limit distribution involves the inverse of the derivative applied to the system of equations. Whereas in the finite-dimensional situation the use of empirical processes can be avoided through higher order Taylor expansions, now empirical processes appear indispensable. But we do not mind that, of course, having established already all the tools we need. For each θ in a subset Θ of a Banach space and each h in an arbitrary set H, let ψθ,h : X → R be a measurable function. Denote by ψθ (x) the vectorvalued function {ψθ,h (x): h ∈ H} and let Pn ψθ and P ψθ be the corresponding vector-valued empirical and “true” means. We are interested in zeros θˆ of the map θ → Pn ψθ . Equivalently, in random elements θˆ with values in Θ such that Pn ψθ,h = 0, every h ∈ H. We expect that the sequence θˆn converges in probability to a zero of the map θ → P ψθ . In applications where θˆ is a maximum likelihood or another contrast estimator, we usually already know this from applying a standard method to the contrast function. It may also be possible to establish consistency from the fact that θˆ is a zero only. In any case, the consistency

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issue does not yield structurally different questions from before and we omit further discussion. We assume that the maps h → ψθ (x) and h → P ψθ,h are uniformly bounded, so that the maps θ → Pn ψθ and θ → P ψθ map Θ into ∞ (H). This may seem a bit special, but even when considering maps θ → ψθ (x) with values in a general Banach space, we can always force this in the present form by choosing the right index set H. The advantage of the present special form is that we set the theorems immediately within the context of empirical processes. √ The following theorem establishes the asymptotic normality of n(θˆ − θ) and should be compared to Theorem 6.17. Recall that Fr´echet differentiability is ordinary differentiability. Thus the map θ → P ψθ is Fr´echet differentiable at θ0 if there exists a continuous, linear map V : lin Θ → ∞ (H) such that, as θ → θ0 ,     P ψθ − P ψθ − V (θ − θ0 ) = o θ − θ0  . 0 H In our setting we do not assume that the domain of the map θ → P ψθ contains θ0 as an interior point, but allow Θ to be arbitrary. The sequence θ → θ0 in the preceding display is restricted to Θ. Theorem 10.1. Suppose that the class of functions {ψθ,h : θ ∈ Θ, h ∈ H} is P -Donsker, that the map θ → P ψθ is Fr´echet differentiable at θ0 with derivative V : lin Θ → ∞ (H) that is one-to-one and has a continuous inverse V −1 : R(V ) → lin Θ. Furthermore, assume that the maps θ → ψθ,h are continuous in L2 (P ) at θ0 , uniformly in h ∈ H. Then any zero θˆn of θ → Pn ψθ that converges in probability to a zero θ0 of θ → P ψθ satisfies √ V n(θˆ − θ0 ) = Gn ψθ0 + oP (1). P 0 in ∞ (H). Equip Proof. The first step is to prove that Gn (ψθˆn − ψθ0 ) → the set H × Θ with the semi-metric    ρ (h, θ), (h , θ ) = P (ψθ,h − ψθ ,h )2 ,

and define a map φ: ∞ (H×Θ)×Θ → ∞  (H) by φ(z, θ) = z(·, θ)−z(·, θ0 ). By  assumption we have that ρ (h, θ), (h, θ0 ) → 0, uniformly in h ∈ H, as θ → θ0 . Thus if z ∈ ∞ (H×Θ) is ρ-uniformly continuous, then |z(h, θ)−z(h, θ0 )| → 0, uniformly in h ∈ H, if θ → θ0 . Consequently, for such z and for (zn , θn ) → (z, θ0 ) an arbitrary sequence in ∞ (H × Θ) × Θ,    φ(zn , θn ) − φ(z, θ0 ) = zn (h, θn ) − zn (h, θ0 )H H   ≤ 2zn − zH×Θ + z(h, θn ) − z(h, θ0 ) → 0. H

We conclude that the map φ is continuous at every point (z, θ0 ) such z is ρ-uniformly continuous at θ0 . Almost all sample paths of a Brownian bridge are uniformly continuous relative to the L2 (P )-norm and therefore almost all

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sample paths (θ, h) → Z(θ, h) of the process Z(θ, η) = Gψθ,h are uniformly continuous relative to ρ. By assumption we have that Zn
Z and that P θˆn → θ0 . Hence (Zn , θˆn )
(Z, θ0 ) and by the continuous mapping theorem we conclude that φ(Zn , θˆn )
φ(Z, θ0 ) = 0. This is equivalent to the claim P that Gn (ψθˆn − ψθ0 ) → 0 in ∞ (H). Using the fact that θˆ and θ0 are zeros we can rewrite the claim as √ − nP (ψθˆn − ψθ0 ) = Gn ψθ0 + oP (1). The right side converges in distribution in ∞ (H), by the Donsker assumption. Hence its norm is OP (1). The left side can be written as  √  − n V (θˆn − θ0 ) + oP (θˆn − θ0 ) by the assumption of Fr´echet differentiability. Because V has a continuous inverse on its range, there exists a constant c > 0 such that V (θ − θ0 ) ≥ cθ − θ0  for every θ ∈ Θ. We use this and the preceding displays to conclude √ that nθˆn − θ0  = OP (1). Next we insert this in the preceding display to √ see that the display is equivalent to −V n(θˆn − θ0 ) + oP (1).

√ We can invert the assertion of the preceding theorem to see that n(θˆn − θ0 ) is asymptotically distributed as V −1 Gψθ0 provided we use the correct (continuous) extension of the inverse operator V −1 to a domain that contains the support of the Brownian bridge Gψθ0 . Because continuous, linear transformations of Gaussian processes are Gaussian we obtain that the se√ quence n(θˆn −θ0 ) is asymptotically normal. In many situations, though, the limit distribution is easier found by performing the inversion of the relation for a finite n. We shall see an example of this in the following treatment of maximum likelihood estimators. An important condition in the theorem is the continuous invertibility of the derivative V . Since a linear map between Euclidean spaces is automatically continuous, in the finite-dimensional set-up this condition reduces to the derivative being one-to-one. For infinite-dimensional systems of estimating equations, the continuity is far from automatic and may be the condition that is hardest to verify. Since it refers to the ∞ (H)-norm, we have some control over it while setting up the system of estimating equations and choosing the set of functions H. A bigger set H makes V −1 more readily continuous, but makes the differentiability of the map θ → P ψθ and the Donsker condition more stringent.

10.2 Maximum Likelihood Consider a semiparametric model, indexed by a finite-dimensional parameter θ of interest and a nuisance parameter η, assumed to be contained in

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some Banach space. We wish to apply the preceding theorem to derive the ˆ ηˆ) of maximum likelihood estimators. asymptotic distribution of the pair (θ, (Thus θ of the theorem becomes (θ, η) in this section.) This approach gives an alternative to the one of Lecture 9 based on the efficient score equation. √A limitation of the present approach is that both θˆ and ηˆ must converge at nrate. It is not clear that a model can always appropriately be parametrized such that this is the case, while it is certainly not always the case for the natural parametrization. An advantage is that we obtain the joint asymptotic distribution of θˆ and ηˆ. The system of estimating equations that we are looking for will consist of stationary equations resulting from varying either the parameter θ or the ˆ ηˆ) nuisance parameter η. Suppose that our maximum likelihood estimator (θ, maximizes the function  (θ, η) → lik(θ, η)(Xi ), for lik(θ, η)(x) being the “likelihood” given one observation x. The parameter θ can be varied in the usual way, and the resulting stationary equation takes the form Pn ˙θ,ˆ ˆ η = 0. This is the usual maximum likelihood equation, except that we evaluate the ˆ ηˆ), rather than at the single value θ. ˆ score function at the joint estimator (θ, A precise condition for this equation to be valid is that the partial derivative of log lik(θ, η)(x) with respect to θ exists and is equal to ˙θ,η (x), for every x, ˆ (at least for η = ηˆ and at θ = θ). Varying the nuisance parameter η is conceptually more difficult. Typically, we can use a selection of the submodels t → ηt used for defining the tangent set and the information in the model. If scores for η take the form of an “operator” Bθ,η working on a set of indices h, then a typical likelihood equation will take the form Pn Bθ,ˆ ˆ η h = Pθ,ˆ ˆ η Bθ,ˆ ˆ η h. Here we have made it explicit in our notation that a score function always has mean zero, by writing the score function as x → Bθ,η h(x) − Pθ,η Bθ,η h rather than as x → Bθ,η h(x). The preceding display will be valid if, for every (θ, η), there exists some path t → ηt (θ, η) such that η0 (θ, η) = η and, for every x, Bθ,η h(x) − Pθ,η Bθ,η h =

  ∂ log lik θ + t, ηt (θ, η) . ∂t |t=0

Assume that this is the case for every h in some index set H, and suppose that the latter is chosen in such a way that the map h → Bθ,η h(x)−Pθ,η Bθ,η h is uniformly bounded on H, for every x and every (θ, η).

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Our total set of estimating equations may be thought of as indexed by the set {1, . . . , k} ∪ H. We can summarize the estimating equations in a random map Ψn : Rk × H → Rk × ∞ (H) given by Ψn = (Ψn1 , Ψn2 ) with Ψn1 (θ, η) = Pn ˙θ,η , Ψn2 (θ, η)h = Pn Bθ,η h − Pθ,η Bθ,η h,

h ∈ H.

The expectation of these maps under the parameter (θ0 , η0 ) is the deterministic map Ψ = (Ψ1 , Ψ2 ) given by Ψ1 (θ, η) = Pθ0 ,η0 ˙θ,η , Ψ2 (θ, η)h = Pθ0 ,η0 Bθ,η h − Pθ,η Bθ,η h,

h ∈ H.

By construction, the maximum likelihood estimators (θˆn , ηˆn ) and the “true” parameter (θ0 , η0 ) are zeros of these maps, Ψn (θˆn , ηˆn ) = 0 = Ψ (θ0 , η0 ). Under some conditions, Theorem 10.1 gives the asymptotic distribution of √ ˙ the sequence n(θˆ − θ0 , ηˆ − η0 ) as √ a function of the derivative Ψ0 of Ψ at (θ0 , η0 ) and the limit process of n(Ψn − Ψ )(θ0 , η0 ), a pair of a Gaussian vector and a Brownian bridge process. We would like to make this limit process more concrete and ascertain that the maximum likelihood estimator is asymptotically efficient. Then we need to relate the derivative of Ψ to the score and information operators of the model. Consider the case that η is a measure on a measurable space (Z, C). Then the directions h can often be taken equal to bounded functions h: Z → R, corresponding to the paths dηt =  (1 + th) dη if η is a completely unknown measure, or dηt = 1 + t(h − ηh) dη if the total mass of each η is fixed to one. In the remainder of the discussion, we assume the second. Now the derivative map Ψ˙ 0 typically takes the form    Ψ˙ Ψ˙ θ − θ0 (θ − θ0 , η − η0 ) → ˙ 11 ˙ 12 , η − η0 Ψ21 Ψ22 where

Ψ˙ 11 (θ − θ0 ) = −Pθ0 ,η0 ˙θ0 ,η0 ˙Tθ0 ,η0 (θ − θ0 ),  Ψ˙ 12 (η − η0 ) = − Bθ∗0 ,η0 ˙θ0 ,η0 d(η − η0 ), Ψ˙ 21 (θ − θ0 )h = −Pθ0 ,η0 (Bθ0 ,η0 h)˙Tθ0 ,η0 (θ − θ0 ),  ˙ Ψ22 (η − η0 )h = − Bθ∗0 ,η0 Bθ0 ,η0 h d(η − η0 ).

(10.1)

For instance, to find the last identity in an informal manner, consider a path ηt in the direction of g, so that dηt −dη0 = tg dη0 +o(t). Then by the definition of a derivative

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  Ψ2 (θ0 , ηt ) − Ψ2 (θ0 , η0 ) ≈ Ψ˙ 22 ηt − η0 + o(t). On the other hand, by the definition of Ψ , for every h, Ψ2 (θ0 , ηt )h − Ψ2 (θ0 , η0 )h = −(Pθ0 ,ηt − Pθ0 ,η0 )Bθ0 ,ηt h ≈ −tPθ0 ,η0 (Bθ0 ,η0 g)(Bθ0 ,η0 h) + o(t)  = − (Bθ∗0 ,η0 Bθ0 ,η0 h) tg dη0 + o(t)  ≈ − (Bθ∗0 ,η0 Bθ0 ,η0 h) d(ηt − η0 ) + o(t). On comparing the preceding pair of displays, we obtain the last line of (10.1). These arguments are purely heuristic, and this form of the derivative must be established for every example. For instance, within the context of Theorem 10.1, we may need to apply Ψ˙ 0 to η that are not absolutely continuous with respect to η0 . Then the validity of (10.1) already depends on the version that is used to define the adjoint operator Bθ∗0 ,η0 . By definition, an adjoint is an operator between L2 -spaces and hence maps equivalence classes into equivalence classes. The continuous invertibility of Ψ˙ 0 can be verified by ascertaining the con−1 ˙ Ψ12 . In tinuous invertibility of the two operators Ψ˙ 11 and V˙ = Ψ˙ 22 − Ψ˙ 21 Ψ˙ 11 that case we have  −1  −1 ˙ −1 −1 Ψ˙ 11 + Ψ˙ 11 Ψ12 V˙ −1 Ψ˙ 21 Ψ˙ 11 −Ψ˙ 11 Ψ12 V˙ −1 −1 ˙ Ψ0 = . −1 V˙ −1 −V˙ −1 Ψ˙ 21 Ψ˙ 11 The operator Ψ˙ 11 is the Fisher information matrix for θ when η is known. If this would not be invertible, then there would be no hope of finding asymptotically normal estimators for θ. The operator V˙ has the form  V˙ (η − η0 )h = − (Bθ∗0 ,η0 Bθ0 ,η0 + K)h d(η − η0 ), where the operator K is defined as   Kh = − Pθ0 ,η0 (Bθ0 ,η0 h)˙Tθ0 ,η0 Iθ−1 Bθ∗0 ,η0 ˙θ0 ,η0 . 0 ,η0 The operator V˙ : lin H → ∞ (H) is certainly continuously invertible if there exists a positive number  such that, for all η ∈ lin H   sup V˙ (η − η0 )h ≥ εη − η0 . h∈H ∞ In the case that η is identified  map h → ηh in  (H), the norm on  with the the right is given by suph∈H (η − η0 )h. Then the display is certainly satisfied if, for some ε > 0,   (Bθ∗0 ,η0 Bθ0 ,η0 + K)h: h ∈ H ⊃ H.

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10 Lecture: Infinite-dimensional Z-Estimators

This condition has a nice interpretation if H is equal to the unit ball of a Banach space D of functions. Then the preceding display is implied by the operator Bθ∗0 ,η0 Bθ0 ,η0 + K: D → D being continuously invertible. The first part of this operator is the information operator for the nuisance parameter. Typically, this would be continuously invertible if the nuisance parameter √ is regularly estimable at a n-rate (relatively to the norm used) when θ is known. The following lemma guarantees that the same is then true for the operator Bθ∗0 ,η0 Bθ0 ,η0 + K if the efficient information matrix for θ is nonsingular, i.e. the parameters θ and η are not locally confounded. Lemma 10.2. Let H be the unit ball in a Banach space D contained in ∞ (Z). If I˜θ0 ,η0 is nonsingular, Bθ∗0 ,η0 Bθ0 ,η0 : D → D is continuous, onto and continuously invertible and Bθ∗0 ,η0 ˙θ0 ,η0 ∈ D, then Bθ∗0 ,η0 Bθ0 ,η0 + K: D → D is continuous, onto and continuously invertible. Proof. Abbreviate the index (θ0 , η0 ) to 0. The operator K is compact, because it has a finite-dimensional range. Therefore, by Lemma 10.3 below, the operator B0∗ B0 + K is continuously invertible provided that it is one-to-one. Suppose that (B0∗ B0 + K)h = 0 for some h ∈ D. By definition Kh = T ∗˙ a0 Bo 0 for for a0 = −I0−1 P0 (B0 h)˙0 . By assumption there exists a path t → ηt with score function B 0 h = B0 h − P0 B0 h at t = 0. Then the submodel indexed by t → (θ0 + ta0 , ηt ) has score function aT0 ˙0 + B 0 h at t = 0, and information aT0 I0 a0 + P0 (B 0 h)2 + 2aT0 P0 ˙0 (B0 h) = P0 (B 0 h)2 + aT0 I0 a0 . Since the efficient information matrix is nonsingular, this information must be strictly positive, unless a0 = 0. On the other hand, 0 = η0 h(B0∗ B0 + K)h = P0 (B0 h)2 − aT0 P0 (B0 h)˙0 . This expression is at least the right side of the preceding display and would be positive if a0 = 0. Thus a0 = 0, whence Kh = 0. Reinserting this in the

equation (B0∗ B0 + K)h = 0, we find that B0∗ B0 h = 0 and hence h = 0. The proof of the preceding lemma is based on the Fredholm theory of linear operators. An operator K: D → D is compact if it maps the unit ball into a totally bounded set. The following lemma shows that for certain operators continuous invertibility is a consequence of their being one-to-one, as is true for matrix operators on Euclidean space. It is also useful to prove the invertibility of the information operator itself. Lemma 10.3. Let D be a Banach space, let the operator A: D → D be continuous, onto and continuously invertible and let K: D → D be a compact operator. Then R(A + K) is closed and has codimension equal to the dimension of N(A + K). In particular, if A + K is one-to-one, then A + K is onto and continuously invertible.

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453

√ The asymptotic covariance matrix of the sequence n(θˆn − θ0 ) can be computed from the expression √ for Ψ˙ 0 and the covariance function of the limiting process of the sequence nΨn (θ0 , η0 ). However, it is easier to use an √ asymptotic representation of n(θˆn − θ0 ) as a sum. For a continuously invertible information operator Bθ∗0 ,η0 Bθ0 ,η0 this can be obtained as follows. In view of (10.1), the assertion of Theorem 10.1 can be rewritten as the system of equations, with a subscript 0 denoting (θ0 , η0 ), √ ηn − η0 )B0∗ ˙0 = −(Pn − P0 )˙0 + oP (1/ n), −I0 (θˆn − θ0 ) − (ˆ √ −P0 (B0 h)˙T (θˆn − θ0 ) − (ˆ ηn − η0 )B ∗ B0 h = −(Pn − P0 )B0 h + oP (1/ n). √

0

0

The oP (1/ n)-term in the second line is valid for every h ∈ H (uniformly in h). If we can also choose h = (B0∗ B0 )−1 B0∗ ˙0 , and subtract the first equation from the second, then we arrive at √ √ I˜θ0 ,η0 n(θˆn − θ0 ) = n(Pn − P0 )˜θ0 ,η0 + oP (1). Here ˜θ0 ,η0 is the efficient score function for θ, as given by Eq. (3.3), and I˜θ0 ,η0 is the efficient information matrix. The representation shows that the √ sequence n(θˆn −θ0 ) is asymptotically linear in the efficient influence function for estimating θ. Hence the maximum likelihood estimator θˆ is asymptotically efficient. Example 10.4 (Cox model). We come back to the Cox model one more time. We recall that the scores and the information operator are given by ˙θ,Λ (x) = δz − zeθz Λ(y)  Bθ,Λ h(x) = δh(y) − eθz

h dΛ

[0,y] ∗ Bθ,Λ Bθ,Λ h(y) = h(y) Eθ,Λ 1Y ≥y eθZ B ∗ ˙θ,Λ = Eθ,Λ 1Y ≥y ZeθZ . θ,Λ

As in the preceding discussion we set up estimating equations Pn ˙θ,Λ = 0 and Pn Bθ,Λ h = 0. Here we let h range over the unit ball of the space BV[0, τ ] of functions h: [0, τ ] → R of bounded variation (with norm the supremum of the uniform norm and the variation norm). The expectations of these equations are given by the maps Ψ1 (θ, Λ) = P0 ˙θ,Λ and Ψ2 (θ, Λ)h = P0 Bθ,Λ h. We can now directly verify the validity of formula (10.1). for the derivative of the map Ψ = (Ψ1 , Ψ2 ) The map Ψ is already linear in Λ. With G0 (y| Z), the distribution function of Y given Z, it can be written as   Ψ1 (θ, Λ) = EZeθ0 Z G0 (y| Z) dΛ0 (y) − EZeθZ Λ(y) dG0 (y| Z),   θ0 Z θZ Ψ2 (θ, Λ)h = Ee h dΛ dG0 (y| Z). h(y)G0 (y| Z) dΛ0 (y) − Ee [0,y]

454

References

The map Ψ : R × ∞ (H) → R × ∞ (H) is linear and continuous in Λ, and its partial derivatives with respect to θ can be found by differentiation under the expectation and are continuous in a neighbourhood of (θ0 , Λ0 ). Several applications of Fubini’s theorem show that indeed the derivative takes the form (10.1). The operator B0∗ B0 , initially introduced as acting on L2 (Λ), can also be viewed as an operator of the space BV[0, τ ] into itself. It is continuously invertible if the function y → Eθ0 ,Λ0 1Y ≥y eθ0 Z is bounded away from zero on [0, τ ], which is part of our assumptions. In Lecture 3 we already computed the efficient information and noted its positivity (under the assumption that Z is not almost surely equal to a function of h(Y )). Thus, we can conclude that the map Ψ˙ 0 is continuously invertible by Lemma 10.2. The class H is a universal Donsker class and hence the first parts δh(y) of the functions Bθ,Λ h form a Donsker class. The functions of the form
h dΛ with h ranging over H and Λ ranging over a collection of measures [0,y] of uniformly bounded variation are functions of uniformly bounded variation and hence also belong to a Donsker class. Thus the functions Bθ,Λ h form a Donsker class by Theorem 6.10. The other conditions of Theorem 10.1 are satisfied too. We finish our lectures with the conclusion that the maximum likelihood estimator in the Cox model, alias the partial likelihood estimator, is asymptotically efficient. We are not the first to conclude this, but we still feel that this is a worthy conclusion of the lectures, remembering that the present approach also applies to other models. Notes. This lecture has its roots in [39].

References

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37. van der Vaart, A.W. (1991): On differentiable functionals. Annals of Statistics 19, 178–204. 38. van der Vaart, A.W. (1991): An asymptotic representation theorem. International Statistical Review97–121. 39. van der Vaart, A.W. (1994: Maximum likelihood estimation with partially censored observations. Annals of Statistics 22, 1896–1916. 40. van der Vaart, A.W. (1996): Efficient estimation in semiparametric models. Annals of Statistics 24, 862–878. 41. van der Vaart, A.W., Wellner, J.A. (1996): Weak Convergence and Empirical Processes New York: Springer. 42. van der Vaart, A.W. (1998): Asymptotic Statistics. Cambridge: Cambridge University Press

LIST OF OTHER TALKS

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LIST OF PARTICIPANTS

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LIST OF PREVIOUS VOLUMES OF THE “Ecole d’Et´ e de Probabilit´ es”

LIST OF PREVIOUS VOLUMES OF THE “Ecole d’Et´e de Probabilit´es”

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LIST OF PREVIOUS VOLUMES OF THE “Ecole d’Et´e de Probabilit´es”

LIST OF PREVIOUS VOLUMES OF THE “Ecole d’Et´e de Probabilit´es”

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LIST OF PREVIOUS VOLUMES OF THE “Ecole d’Et´e de Probabilit´es”